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OSMANIA UNiyERSItY LIBRARY

Call No. 62~*?*/3i~~]> // v//s Accession No,
Author ^

Title A ^f\J^ , * .

/HSl+g*^^

This book sli I re the date Kst marked below.

AERODYNAMICS

By the Same Author

A COMPLETE COURSE IN ELEMENTARY
AERODYNAMICS WITH EXPERIMENTS
AND EXAMPLES

AERODYNAMICS

N. A. V. PIERCY

D.Sc., M.Inst.C.E., M.I.Mech.E., F.R. Ae.S.

Reader in Aeronautics in the University of London

Head of the Department of Aeronautics, Queen Mary College

Member of the Association of Consulting Engineers

SECOND EDITION

THE ENGLISH UNIVERSITIES PRESS LTD
LONDON

FIBST PRINTED 1937

REPRINTED ...... 1943

SECOND EDITION, REVISED AND ENLAKGED 1947

Made and Printed in Great Britain by
Hagell, Watson 6* Viney Ltd., London and Aylesbury.

PREFACE TO SECOND EDITION

THE present edition is enlarged to provide, in the first place, an
introduction to the mathematical and experimental study of com-
pressible flow, subsonic and supersonic. This and other matters
now becoming prominent are not collected in a supplementary
section but incorporated in place as additional articles or short
chapters. Following a well-established practice, the numbering
of original articles, figures and chapters is left undisturbed as far
as possible, interpolations being distinguished by letter-suffixes.
It is hoped this procedure will ensure a minimum of inconvenience
to readers familiar with the earlier edition. To some extent the
unlettered articles indicate a first course of reading, though a modern
view of Aerodynamics requires consideration of Mach numbers
equally with Reynolds numbers almost from the outset.

Other matters now represented include various theories of thin
aerofoils and the reduction of profile drag. The brief account of
the laminar-flow wing is in general terms, but the author has drawn
for illustrations on the conformal system, in the development of
which he has shared more particularly.

The original text is revised to bring it up to date, and also in the
following connection. Experience incidental to the use of the book
at Cambridge and London Universities isolated certain parts where
the treatment was insufficiently detailed for undergraduates ; these
are now suitably expanded.

The aim of the book remains unchanged. It does not set out to
collect and summarise the researches, test results and current
practice of the subject, but rather to provide an adequate and
educational introduction to a vast specialist literature in a form that
will be serviceable for first and higher degrees, and like purposes,
including those of the professional engineer.

N. A. V. PIERCY.

TEMPLE,

October, 1946.

VI PREFACE

PREFACE TO FIRST EDITION

FIRST steps towards formulating the science of Aerodynamics pre-
ceded by only a few years the epoch-making flight by the Wright
brothers in 1903. Within a decade, many fundamentals had been
established, notably by Lanchester, Prandtl, Joukowski, and Bryan.
Yet some time elapsed before these essentially mathematical con-
ceptions, apart from aircraft stability, were generally adopted.
Meanwhile, development proceeded largely by model experiment.
To-day, much resulting empiricism has been superseded and the
subject is unique among those within the purview of Engineering in
its constant appeal to such masters as Helmholtz and Kelvin,
Reynolds and Rankine. A complete theory is stiU far out of
reach ; experiment, if no longer paramount, remains as important
as analysis ; and there is a continual swinging of the pendulum
between these two, with progress in aviation marking time.

This book presents the modern science of Aerodynamics and its
immediate application to aircraft. The arrangement is based on
some eighteen years' organisation of teaching and research in the
University of London. The first five chapters, and the simpler
parts of Chapters VI-XII, constitute an undergraduate course ; more
advanced matters are included to serve especially the Designer and
Research Engineer. No attempt has been made to summarise
reports from the various Aerodynamic Laboratories, which must be
consulted for design data, but the treatment is intended to provide
an adequate introduction to the extensive libraries of important
original papers that now exist in this country and abroad.

To facilitate reference, symbols have been retained, for the most
part, in familiar connections, though duplication results in several
instances, as shown in the list of notations. Of the two current
systems of force and moment coefficients, the American or Contin-
ental, associated with " C," will probably supersede the British,
distinguished by " k." No great matter is involved, a C-coefficient
being derived merely by doubling the corresponding ^-coefficient.
However, so many references will be made in this country to litera-
ture using the ' k" notation that the latter has been given some
preference.

My thanks are due to Professor W. G. Bickley for reading the
proof sheets and making many suggestions ; also to The English
Universities Press for unremitting care and consideration.

N. A. V. PIERCY.
TEMPLE.

ART.

PREFACE
NOTATION

CONTENTS

CHAPTER I

PAGE
V

xiii

AIR AT REST, THE ATMOSPHERE AND STATIC LIFT

1—4. Properties of Air ; Density ; Pressure ....

5—7. Hydrostatic Equation. Incompressibility Assumption.
Measurement of Small Pressures .....

8-9. Buoyancy of Gas-filled Envelope. Balloons and Airships .
10. Centre of Pressure ........

11—15. Relation between Pressure, Density, and Temperature of a
Gas. Isothermal Atmosphere. Troposphere. The In-
ternational Standard Atmosphere. Application to Alti-
meters .........

16-17. Gas-bag Lift in General. Vertical Stability

18. Atmospheric Stability and Potential Temperature
19-20. Bulk Elasticity. Velocity of Sound

CHAPTER II
AIR FLOW AND AERODYNAMIC FORCE

21. Streamlines and Types of Flow .....

22. Absence of Slip at a Material Boundary ....
23-25. Viscosity : Qualitative Theory ; Maxwell's Definition ; Ex-
perimental Laws ........

26-28. Relation between Component Stresses in Non-uniform Flow ;
Static Pressure. Forces on an Element

29-33. Bernoulli's Equation : Variation of Density and Pressure ;
Adiabatic Flow ; Temperature Variation ; the Incompres-
sible Flow Assumption ; Pitot Tube ; Basis of Velocity
Measurement ........

34-41. Equation of Continuity. Experimental Streamlines. Stream
Function. Circulation and Vorticity. Gradient of Pitot
Head across Streamlines. Irrotational Flow .

42-43. The Boundary Layer Experimentally Considered

44-46. Constituents of Aerodynamic Force. Integration of Normal
Pressure and Skin Friction ......

47-49c. Rayleigh's Formula. Reynolds Number. Simple Dynami-
cally Similar Motions. Aerodynamic Scale. Mach Num-
ber. Froude Number. Corresponding Speeds

CHAPTER III

WIND-TUNNEL EXPERIMENT

50-53. Nature of Wind-tunnel Work. Atmospheric Tunnels

54. Coefficients of Lift, Drag, and Moment ....

vii

13
18
20
21

23
26

26
32

44
52

68
75

Vlll

CONTENTS

ART.
66-69.

69A.
60.
61-63.

64-66.

Suspension of Models. Double Balance Method. Aero-
dynamic Balance. Some Tunnel Corrections

Pitot Traverse Method

Aerofoil Characteristics .......

Application of Complete Model Data ; Examples. Arrange-
ment of Single Model Experiment. Compressed-air
Tunnel

Practical Aspect of Aerodynamic Scale. Scale Effects.
Gauge of Turbulence .......

PAGE

77
86
89

CHAPTER III A
EXPERIMENT AT HIGH SPEEDS

66. Variable-density Tunnel 103

66A. Induced-flow Subsonic Tunnel. Wall Adjustment.

Blockage 106

66B-66C. Supersonic Tunnel. Illustrative Results . . . .109
66D. Pitot Tube at Supersonic Speeds. Plane Shock Wave . 114

CHAPTER IV
AIRCRAFT IN STEADY FLIGHT

67-69. Examples of Heavier-than-air Craft. Aeroplanes v. Air-
ships. Aeroplane Speed for Minimum Drag .

70-72. Airship in Straight Horizontal Flight and Climb

73-76. Aeroplane in Level Flight. Size of Wings; Landing Con-
ditions ; Flaps . . . . . ...

77-79. Power Curves ; Top Speed ; Rate of Climb

80. Climbing, Correction for Speed .....

81-83. Effects of Altitude, Loading, and Partial Engine Failure .

84-85. Gliding; Effects of Wind; Motor-less Gliders

86-89. Downwash. Elevator Angle; Examples; C.G. Location.
90. Nose Dive

91-96. Circling and Helical Flight. Rolling and Autorotation.
Handley Page Slot. Dihedral Angle ....

118
126

129
137
140
143
146
149
155

156

CHAPTER V
FUNDAMENTALS OF THE IRROTATIONAL FLOW

96-100.
101-106.
106-109.

110-114.
115.

116-117.
118-120.

Boundary Condition. Velocity-potential. Physical Mean-
ing of <f>. Potential Flow. Laplace's Equation

Source. Sink. Irrotational Circulation. Combined Source
and Sink. Doublet .....

Flow over Faired Nose of Long Board. Oval Cylinder.
Circular Cylinder without and with Circulation

Potential Function. Examples. Formulae for Velocity

Circulation round Elliptic Cylinder or Plate. Flow through
Hyperbolic Channel .......

Rankine's Method. Elliptic Cylinder or Plate in Motion .

Acceleration from Rest. Impulse and Kinetic Energy of
the Flow Generated by a Normal Plate ....

163
167

172

180

184
186

190

CONTENTS

CHAPTER VI
TWO-DIMENSIONAL AEROFOILS

ART.

121-124.

PAGE

Conformal Transformation; Singular Points. Flow past
Normal Plate by Transformation. Inclined Plate . .194

125-127. Joukowski Symmetrical Sections ; Formulae for Shape.
Velocity and Pressure. More General Transformation
Formula. Karman-Trefftz Sections .... 203

128-129B. Piercy Symmetrical Sections. Approximate Formulae.
Velocity over Profile. Comparison with Experiment and
Example 212

130-133A. Circular Arc Aerofoil. Joukowski and Piercy Wing Sections 220

134-139. Joukowski's Hypothesis ; Calculation of Circulation ; Stream-
lines with and without Circulation. Investigation of Lift,

Lift Curve Slope and Moment 227

140. Comparison with Experiment ..... 235

CHAPTER VIA
THIN AEROFOILS AT ORDINARY SPEEDS

140A-140B. Method and Equations .

140c. Application to Circular Arc
140D-140F. General Case. Aerodynamic Centre.

Example

237
240
241

CHAPTER VIB
COMPRESSIBLE INVISCID FLOW

140G-140i. Assumptions. General Equation of Continuity , . 245
140J-140K. Euler's Dynamical Equations. Kelvin's (or Thomson's)

Theorem 247

140L-140M. Irrotational Flow. Integration of Euler's Equations . 260
140N-140O. Steady Irrotational Flow in Two Dimensions. Electrical

and Hydraulic Analogies ...... 252

CHAPTER VIC

THIN AEROFOILS AT HIGH SPEEDS

140P-140Q. Subsonic Speeds. Glauert's Theory. Comparison with

Experiment. Shock Stall 259

140R-140T. Supersonic Speeds. Mach Angle. Ackeret's Theory. Com-
parison with Experiment . . . . .262

CHAPTER VII
VORTICES AND THEIR RELATION TO DRAG AND LIFT

141-147. Definitions. Rankine's Vortex. General Theorems . 267

148-150. Induced Velocity for Short Straight Vortex and Vortex

Pair. Analogies 272

X CONTENTS

ART. PAGE

151-155. Constraint of Walls. Method of Images ; Vortex and Vor-
tex Pair within Circular Tunnel; Other Examples. Ap-
plication of Conformal Transformation; Streamlines for
Vortex between Parallel Walls . . . . .276

155A-155B.Lift from Wall Pressures. Source and Doublet in Stream

between Walls ........ 283

156-162. Generation of Vortices ; Impulse ; Production and Dis-
integration of Vortex Sheets. Karman Trail ; Applica-
tion to Circular Cylinder. Form Drag .... 287

163-168. Lanchester's Trailing Vortices. Starting Vortex. Residual
Kinetic Energy ; Induced Drag ; Example of Uniform
Lift. Variation of Circulation in Free Flight. Example
from Experiment ....... 295

CHAPTER VIII

WING THEORY

169-171. General Equations of Monoplane Theory .... 309

172-177. The 'Second Problem.' Distribution of Given Impulse for
Minimum Kinetic Energy ; Elliptic Loading. Minimum
Drag Reduction Formulae ; Examples . . .312

178-180. Solution of the Arbitrary Wing by Fourier Series. Elliptic
Shape Compared with Others. Comparison with Experi-
ment ......... 320

181-186. General Theorems Relating to Biplanes. Prandtl's Biplane
Factor ; Examples. Equal Wing Biplane — Comparison
with Monoplane ; Examples ..... 327

187-188. Tunnel Corrections for Incidence and Induced Drag . . 335
189-192. Approximate Calculation of Downwash at Tail Plane ; Tun-
nel Constraint at Tail Plane ; Correction Formulae. Tail
Planes of Biplanes 339

CHAPTER IX
VISCOUS FLOW AND SKIN DRAG

193-199. Laminar Pipe Flow : Theory and Comparison with Experi-
ment. Turbulent Flow in Pipes ; the Seventh-root Law.
Flow in Annular Channel. Eccentric and Flat Cores in

Pipes 346

200-204. General Equations for Steady Viscous Flow. Extension of

Skin Friction Formula . . . . . .357

205-207. Viscous Circulation. Stability of Curved Flow . . . 365
208-209. Oseen's and Prandtl's Approximate Equations . . . 369
210-2 1 7 . Flat Plates with Steady Flow : Solutions for Small and Large
Scales ; Formation of Boundary Layer ; Method of Suc-
cessive Approximation. Karman's Theorem ; Examples 370
2 18-2 ISA. Transition Reynolds Number. Detection of Transition . 384
219-221. Flat Plates with Turbulent Boundary Layers : Power

Formulae. Transitional Friction. Experimental Results 387
221A-221B. Displacement and Momentum Thicknesses. Alternative

Form of Kdrm&n's Equation . . . . .391

CONTENTS

ART.

222-223.

224-230.

PAGE

Note on Laminar Skin Friction of Cylindrical Profiles.
Breakaway. Effect of Wake. Frictions of Bodies and
Flat Plates Compared 393

Turbulence and Roughness. Reynolds Equations of Mean
Motion. Eddy Viscosity. Mixing Length. Similarity
Theory. Skin Drag. Application to Aircraft Surfaces.
Review of Passage from Model to Full Scale . . . 399

CHAPTER IX A
REDUCTION OF PROFILE DRAG

230A-230B. Normal Profile Drag. Dependence of Friction on Transition
Point

230c-230F. Laminar Flow Wings. Early Example. Maintenance of
Negative Pressure Gradient. Position of Maximum
Thickness. Incidence Effect ; Favourable Range. Veloc-
ity Diagrams. Examples of Shape Adjustment. Camber
and Pitching Moment .......

230G-230H. Boundary Layer Control. Cascade Wing

230i. Prediction of Lift with Laminar Boundary Layer .

230j~230K. High Speeds,
back

Minimum Maximum Velocity Ratio. Sweep-

409

412
419
421

423

CHAPTER X
AIRSCREWS AND THE AUTOGYRO

231-232. The Ideal Propeller ; Ideal Efficiency of Propulsion . . 425

233-238. Airscrews. Definitions. Blade Element Theory. Vortex
Theory ; Interference Factors ; Coefficients ; Method of
Calculation ; Example ...... 427

239. Variable Pitch. Static Thrust 438

240-241. Tip Losses and Solidity. Compressibility Stall . . 440

242. Preliminary Design : Empirical Formulae for Diameter and

Inflow ; Shape ; Stresses ...... 443

243-245. Helicopter and Autogyro. Approximate Theory of Auto-
gyro Rotor. Typical Experimental Results . . , 446

CHAPTER XI
PERFORMANCE AND EFFICIENCY

246-260. Preliminary Discussion. Equivalent Monoplane Aspect

Ratio. Induced, Profile and Parasite Drags; Examples 451
251. Struts and Streamline Wires . . . . . .457

252-263. Jones Efficiency ; Streamline Aeroplane. Subdivision of

Parasite Drag ........ 469

254-258. Airscrew Interference ; Example . . . . .461

259-260A. Prediction of Speed and Climb ; Bairstow's and the Lesley-
Reid Methods. Method for Isolated Question . . 466

261-262. Take-off and Landing Run. Range and Endurance. , 473

Xll

CONTENTS

ART.

PAGE

262A-262F. Aerodynamic Efficiency ; Charts. Airscrew Effects ; Ap-
plication to Prediction ; Wing-loading and High-altitude

Flying ; Laminar Flow Effect 477

263. Autogyro and Helicopter ...... 487

263A. Correction of Flight Observations ..... 489

CHAPTER XII

SAFETY IN FLIGHT

264-265. General Problem. Wind Axes. Damping Factor . .493
266-269. Introduction to Longitudinal Stability : Aerodynamic Dihe-
dral ; Short Oscillation ; Examples ; Simplified Phugoid
Oscillation ; Example 496

270-273. Classical Equations for Longitudinal Stability. Glauert's
Non-dimensional System. Recast Equations. Approxi-
mate Factorisation ....... 503

274-278. Engine-off Stability : Force and Moment Derivatives.

Example. Phugoid Oscillation Reconsidered . . 508

279-280. Effects of Stalling on Tail Efficiency and Damping . . 513
281-284. Level Flight ; High Speeds ; Free Elevators ; Climbing . 514

285. Graphical Analysis . . . . . . .516

286. Introduction to Lateral Stability 618

287-289. Asymmetric Equations ; Solution with Wind Axes ; Ap-
proximate Factorisation . . . . . .519

290-292. Discussion in Terms of Derivatives. Example. Evalua-
tion of Lateral Derivatives . . . . . .521

293-295. Design and Stalling of Controls. Control in Relation to

Stability. Large Disturbances. Flat Spin . . . 523
296. Load Factors in Flight ; Accelerometer Records . . 526

AUTHOR INDEX 629

SUBJECT INDEX

531

NOTATION

(Some of the symbols are also used occasionally in connections
other than those stated below.)

A . . . Aerodynamic force ; aspect ratio ; transverse
moment of inertia.

A.R.C.R. & M. Aeronautical Research Committee's Reports and
Memoranda.

A.S.I. . . Air speed indicator.

a . . . Axial inflow factor of airscrews ; leverage of
Aerodynamic force about C.G. of craft ; slope
of lift curve of wings ; velocity of sound in air.

a9 . . . Slope of lift curve of tail plane.

a . . . Angle of incidence.

a, . . . Tail-setting angle.

B . . . Gas constant ; longitudinal moment of inertia ;
number of blades of an airscrew.

Blt B* . . Stability coefficients.

b . . . Rotational interference factor of airscrews.

fi . . . Transverse dihedral ; twice the mean camber of a
wing.

C . . . Directional moment of inertia ; sectional area of
tunnel.

C.A.T. . . Compressed air tunnel.

Clf C2 . . Stability coefficients.

CL, CD, etc. . Non-dimensional coefficients of lift, drag, etc., on
basis of stagnation pressure.

C.G. . . Centre of gravity.

C.P. . . Centre of pressure.

c . . . Chord of wing or aerofoil ; molecular velocity.

y . . . Ratio of specific heats ; tan'1 (drag/lift).

D . . . Diameter ; drag.

Dlt D2 . . Stability coefficients.

A, 8 . . Thickness of boundary layer; displacement thick-
ness.

E . . . Elasticity ; kinetic energy.

Elt Et . . Stability coefficients.

MV NOTATION

e . . . Base of the Napierian logarithms.

e . . . Angle of downwash.

F . . . Skin friction (in Chapter II) ; Froude Number.

~ . . Frequency.

£ . . . Vorticity.

g . . . Acceleration due to gravity.

H . . . Horse power ; the boundary layer ratio 8/0.

h . . . Aerodynamic gap ; height or altitude.

7] . . .A co-ordinate ; efficiency ; elevator angle.

6 . . . Airscrew blade angle ; angle of climb ; angular

co-ordinate ; temperature on the Centigrade

scale ; momentum thickness.

/ . . . Impulse ; second moment of area.
i . . . V-l ; as suffix to D : denoting induced drag ;

incidence of autogyro disc.
/ . . . The advance of an airscrew per revolution in terms

of its diameter.
K . . . Circulation.
k . . . Radius of gyration ; roughness.
^A» ^B» *c • Inertia coefficients.
*L» *D» etc- • Non-dimensional coefficients of lift, drag, etc., on

basis of twice the stagnation pressure.

k99 kx . . British drag and lift coefficients of autogyro rotor.
L . . . Lift ; rolling moment.

/ . . . Length ; leverage of tail lift about C.G. of craft.
X . . . Damping factor ; mean free path of molecule ;

mean lift per unit span.

M . . Pitching moment ; Mach number.

m . . . Mass ; with suffix : non-dimensional moment

derivative ; Mach angle,
ji . . . Coefficient of viscosity ; * relative density of

aeroplane ' ; a co-ordinate.
N . . . Yawing moment.
N.A.C.A. . National Advisory Committee for Aeronautics,

U.S.A.

N.P.L. . . National Physical Laboratory, Teddington.
n . . . Distance along a normal to a surface ; revolutions

per sec.

v . . . Kinematic coefficient of viscosity ; a co-ordinate.
5 • • .A co-ordinate.
P . . . Pitch of an airscrew ; pressure gradient ; total

pressure.

NOTATION XV

p . . . Angular velocity of roll ; pressure or stress.

p . . . Density of air in slugs per cu. ft.

Q . . . Torque.

q . . . Angular velocity of pitch ; resultant fluid

velocity.

R . . . Radius ; Reynolds number.
Rlt Rt . . Routh's discriminant.

R.A.E. . . Royal Aircraft Establishment, Farnborough.
r . . . Angular velocity of yaw ; lift/drag ratio ;

radius.

ra . . . Over-all lift /drag ratio.
5 . . . Area, particularly of wings.

s . . . Distance along contour or streamline ; semi-span,
a . . . Density of air relative to sea-level standard ;

Prandtl's biplane factor ; sectional area of

vortex ; solidity of an airscrew.
r . . . Thickness ; thrust.
t . . . Period of time in sec. ; the complex co-ordinate

5 + «j.

£0 • • • Unit of time in non-dimensional stability equa-
tions.

T . . . Absolute temperature ; skin friction in Chapter
IX ; tail angle of aerofoil section ; tail volume
ratio.

<f> . . . Aerodynamic stagger ; angle of bank ; angle of
helical path of airscrew element ; velocity

'n>n ; yaw.

f undisturbed velocity in the direc-
tOz.
jcity of a body.

lircraft.
/ components in the directions Ox,

Deity ; mean loading of wing in

at ; potential function $ + i^.
it ibrce.
)f stability charts.

'dinates ; with suffixes : non-dimen-

derivatives.

NOTATION

z • • • The complex co-ordinate x + iy.

& • - • Angular velocity of an airscrew.

o> . . . Angular velocity.

t& • - . Impulsive pressure.

V2 • . 9'/9*1 +

V4 . . (

Chapter I
AIR AT REST, THE ATMOSPHERE AND STATIC LIFT

i. Air at sea-level consists by volume of 78 per cent, nitrogen,
21 per cent, oxygen, and nearly 1 per cent, argon, together with
traces of neon, helium, possibly hydrogen, and other gases.
Although the constituent gases are of different densities, the mixture
is maintained practically constant up to altitudes of about 7 miles in
temperate latitudes by circulation due to winds. This lower part of
the atmosphere, varying in thickness from 4 miles at the poles to 9
miles at the equator, is known as the troposphere. Above it is the
stratosphere, a layer where the heavier gases tend to be left at lower
levels until, at great altitudes, such as 50 miles, little but helium or
hydrogen remains. Atmospheric air contains water-vapour in
varying proportion, sometimes exceeding 1 per cent, by weight.

From the point of view of kinetic theory, air at a temperature of
0° C. and at standard barometric pressure (760 mm. of mercury)
may be regarded statistically as composed of discrete molecules, of
mean diameter 1-5 X 10 ~5 mil (one-thousandth inch), to the number
of 4-4 x 1011 per cu. mil. These molecules are moving rectilinearly
in all directions with a mean velocity of 1470 ft. per sec., i.e. one-
third faster than sound in air. They come continually into collision
with one another, the length of the mean free path being 0-0023 mil.

2. Density

Air is thus not a continuum. If it were, the density at a point
would be defined as follows : considering the mass M of a small
volume V of air surrounding the point, the density would be
the limiting ratio of M/V as V vanishes. But we must suppose
that the volume V enclosing the point is contracted only until it is
small compared with the scale of variation of density, while it still
remains large compared with the mean distance separating the
molecules. Clearly, however, V can become very small before the
continuous passage of molecules in all directions across its bounding
surface can make indefinite the number of molecules enclosed and
M or M/V uncertain. Density is thus defined as the ratio of the

A.D.— l 1

2 AERODYNAMICS [CH.

mass of this very small, though finite, volume of air — i.e. of the
aggregate mass of the molecules enclosed — to the volume itself.

Density is denoted by p, and has the dimensions M/Z,8. In
Aerodynamics it is convenient to use the slug-ft.-sec. system of
units.* At 15° C. and standard pressure 1 cu. ft. of dry air weighs
0-0765 Ib. This gives p = 0-0765/g = 0-00238 slug per cu. ft.

It will be necessary to consider in many connections lengths, areas,
and volumes that ultimately become very small. We shall tacitly
assume a restriction to be imposed on such contraction as discussed
above. To take a further example, when physical properties are
attached to a ' point ' we shall have in mind a sphere of very small
but sufficient radius centred at the geometrical point.

3. Pressure

Consider a small rigid surface suspended in a bulk of air at rest.
The molecular motion causes molecules continually to strike, or
tend to strike, the immersed surface, so that a rate of change of
molecular momentum occurs there. This cannot have a component
parallel to the surface, or the condition of rest would be disturbed.
Thus, when the gas is apparently at rest, the aggregate rate of
change of momentum is normal to the surface ; it can be
represented by a force which is everywhere directed at right
angles towards the surface. The intensity of the force per unit
area is the pressure pt sometimes called the hydrostatic or static
pressure.

It is important to note that the lack of a tangential component to
p depends upon the condition of stationary equilibrium. The
converse statement, that fluids at rest cannot withstand a tangential
or shearing force, however small, serves to distinguish liquids from
solids. For gases we must add that a given quantity can expand to
fill a volume, however great.

It will now be shown that the pressure at a point in a fluid at rest is
uniform in all directions. Draw the small tetrahedron ABCO, of

* In this system, the units of length and time are the foot and the second, whilst
forces are in pounds weight. It is usual in Engineering, however, to omit the
word ' weight/ writing * Ib.' for ' lb.-wt.,' and this convention is followed. The
appropriate unit of mass is the 'slug,' viz. the mass of a body weighing g Ib.
Velocities are consistently measured in ft. per sec., and so on. This system being
understood, specification of units will often be omitted from calculations for brevity.
For example, when a particular value of the kinematic viscosity is given as a number,
sq. ft. per sec. will be implied. It will be desirable occasionally to introduce special
units. Thus the size and speed of aircraft are more easily visualised when weights
are expressed in tons and velocities in miles per hour. The special units will be
duly indicated in such cases. Non-dimeasional coefficients are employed wherever
convenient.

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 3

which the faces OAB, OBC, OCA are mutually at right angles (Fig.

1). Denote by S the area of the face ABC. With the help of OD

drawn perpendicular to this face, it is

easily verified from the figure that the

area OCA is S . cos a. The pressure pABC

on the S-face gives rise to a force pAEC-S

which acts parallel to DO. From the

pressures on the other faces, forces simi-

larly arise which are wholly perpendicular

to the respective faces.

Resolving in the direction BO, for
equilibrium
W + pABC . 5 . cos a - pocA . S . cos a = 0, FIG. i.

where W is a force component on the tetrahedron arising from some
general field of force in which the bulk of air may be situated ; such
might be, for example, the gravitational field, when W would be the
weight of the tetrahedron if also OB were vertical. But W is pro-
portional to the volume of the tetrahedron, i.e. to the third order of
small quantities, and is negligible compared with the other terms
of the equation, which are proportional to areas, i.e. to the second
order of small quantities. Hence :

Similarly :

4. It will be of interest to have an expression for p in terms of
molecular motions.

Considering a rigid plane surface suspended in air, draw Oy, Oz
mutually at right angles in its plane and Ox perpendicular to it (Fig.
2). Erect on a unit area S of the plane, and to one side of it a right
cylinder of unit length, so that it encloses unit volume of air. If m is
the mass of each molecule, the total number N of molecules enclosed
is p/w. They are moving in all directions with mean velocity c along
straight paths of mean length X.

At a chosen instant the velocities of all the molecules can be
resolved parallel to Ox, Oy, Oz. But, since N is very large, it is
equivalent to suppose that JV/3 molecules move parallel to each of
the co-ordinate axes with velocity c during the short time A* required
to describe the mean free path. Molecules moving parallel to Oy,
Oz cannot impinge on S ; we need consider only molecules moving

AERODYNAMICS [CH.

parallel to Ox, and of these only
one-half must be taken as moving
towards 5, i.e. in the specific direction
Ox (Fig. 2).

The interval of time corresponding
to the free path is given by —

A* = \/c.

During this interval all those mole-
cules moving in the direction Ox
FIG. 2. which are distant, at the beginning

of A/, no farther than X from S, will

strike S. Their number is evidently >JV/6. Each is assumed
perfectly elastic, and so will have its velocity exactly reversed.
Thus the aggregate change of momentum at S in time A2 is
2mc . AIV/6. The pressure p, representing the rate of change of
momentum, is thus given by :

6. A*

= *?* 0)

Thus the pressure amounts to two-thirds of the molecular kinetic
energy per unit volume.

5. The Hydrostatic Equation

We now approach the problem of the equilibrium of a bulk of air
at rest under the external force of gravity, g has the dimensions of
an acceleration, L/T*. Its value depends slightly on latitude and
altitude, increasing by 0-5 per cent, from the equator to the poles
and decreasing by 0-5 per cent, from sea-level to 10 miles altitude,
At sea-level and 45° latitude its value is 32»173
in ft.-sec. units. The value 32-2 ft./sec.2 is suffi-
ciently accurate for most purposes.

Since no horizontal component of external force
acts anywhere on the bulk of air, the pressure
in every horizontal plane is constant, as otherwise
motion would ensue. Let h represent altitude, so
that it increases upward. Consider an element-
cylinder of the fluid with axis vertical, of length
SA and cross-sectional area A (Fig. 3). The
pressure on its curved surface clearly produces Fia. 3.

^fc-^'^
|pA

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 6

no resultant force or couple. If p is the pressure acting upward
on the lower end of the cylinder, the pressure acting downward

on its upper end will be p + J~8h. These pressures give a resultant

downward force : A-~8h. The gravity force acting on the cylinder
ah

is pg . A8h. Therefore, for equilibrium—

Thus the pressure decreases with increase of altitude at a rate equal
to the local weight of the fluid per unit volume.

6. Incompressibility Assumption in a Static Bulk of Air

Full use of (2) requires a knowledge of the relationship existing
between p and p, but the particular case where p is constant is im-
portant. We then have

ftp = — pg \dh + const.

or for the change between two levels distinguished by the suffixes
1 and 2 :

~hi)- • • • (3)

This equation is exact for liquids, and explains the specification of
a pressure difference by the head of a liquid of known density which
the pressure difference will support. In the mercury barometer, for
example, if A, > hlt p* = 0 and pl is the atmospheric pressure which
supports the otherwise unbalanced cohimn of mercury. At 0° C.
the density of mercury relative to that of water is 13-596. When
A, — A! = 760 mm., pt is found from (3) to be 2115-6 Ib. per sq. ft. at
this temperature.

7. Measurement of Small Pressure Differences

Accurate measurement of small differences of air pressure is often
required in experimental aerodynamics. A convenient instrument
is the Chattock gauge (Fig. 4). The rigid glasswork AB forms a
U-tube, and up to the levels L contains water, which also fills the
central tube T. But above L and the open mouth of T the closed
vessel surrounding this tube is filled with castor oil. Excess of air
pressure in A above that in B tends to transfer water from A to B

6 AERODYNAMICS [CH.

by bubbling through the castor oil. But this is prevented by tilting
the heavy frame F, carrying the U-tube, about its pivots P by means
of the micrometer screw S, the water-oil meniscus M being observed
for accuracy through a microscope attached to F. Thus the excess
air pressure in A is compensated by raising the water level in B
above that in A, although no fluid passes. The wheel W fixed to S
is graduated, and a pressure difference of O0005 in. of water is easily

FIG. 4. — CHATTOCK GAUGE.

detected. By employing wide and accurately made bulbs set close
together, constantly removing slight wear, protecting the liquids
against appreciable temperature changes and plotting the zero
against time to allow for those that remain, the sensitivity * may be
increased five or ten times. These gauges are usually constructed
for a maximum pressure head of about 1 in. of water. Longer forms
extend this range, but other types are used for considerably greater
heads.

At 15° C. 1 cu. ft. of water weighs 62-37 Ib. Saturation with air
decreases this weight by about 0'05 Ib. The decrease of density
from 10° to 20° C. is 0-15 per cent. A 6 or 7 pet cent, saline solution
is commonly used instead of pure water in Chattock gauges, however,
since the meniscus then remains clean for a longer period.

8. Buoyancy of Gas-filled Envelope

The maximum change of height within a balloon or a gas-bag of an
airship is usually sufficiently small for variation of density to be

* Cf. also Cope and Houghton, Jour. Sci. Jnstr., xiii, p. 83, 1936.

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 7

neglected. Draw a vertical cylinder of small cross-sectional area A
completely through the envelope E (Fig. 5), which is filled with a
light gas of density p', and is at rest relative to the surrounding
atmosphere of density p. Let the cylinder cut the envelope at a
lower altitude-level ht and at an upper one Aa, the curves of inter-

FIG. 6.

section enclosing small areas Slt Sa, the normals to which (they are
not necessarily in the same plane) make angles oclf a8 with the
vertical. On these areas pressures />',, p'2t act outwardly due to the
gas, and^>lf p9 act inwardly due to the atmosphere.

There arises at h2 an upward force on the cylinder equal to

(pi — ^a)S2 cos a,.

The similar force arising at h^ may be upward or downward, depend-
ing on the position of Sl and whether an airship or a balloon is
considered, but in any case its upward value is —

(Pi — P()SI cos «i-

Since Sa cos oc8 = A == St cos al, the resultant upward force on the
cylinder due to the pressures is

Substituting from (3), if AL denotes the element of lift —
AL = ~

AERODYNAMICS

[CH. I

The whole volume of the envelope may be built up of a large number
of such cylinders, and its total lift is :

L = (p - p'fcZfA. - hJA

= (P-p')gF' ..... (4)

where V is the volume of gas enclosed. For free equilibrium a
weight of this amount, less the weight of the envelope, must be
attached.

The above result expresses, of course, the Principle of Archimedes.
It will be noted that the lift acts at the centre of gravity of the
enclosed gas or of the air displaced, called the centre of buoyancy, so
that a resultant couple arises only from a displacement of the centre
of buoyancy from the vertical through the centre of gravity of the
attached load plus gas. For stability the latter centre of gravity
must be below the centre of buoyancy.

If W is the total load supported by the gas and a' the density of
the gas relative to that of the surrounding air, (4) gives

W = 9gV'(l-v') (5)

For pure hydrogen, the lightest gas known, a' = 0-0695. But
hydrogen is inflammable when mixed with air and is replaced where
possible by helium, for which a' = 0-138 in the pure state.

9. Balloons and Airships

In balloons and airships the gas is contained within envelopes of
cotton fabric lined with gold-beaters1 skins or rubber impregnated.
Diffusion occurs through these comparatively impervious materials,

and, together with leakage, con-
taminates the enclosed gas, so that
densities greater than those given in
the preceding article must be
assumed.

Practical values for lift per
thousand cubic feet are 68 Ib. for
hydrogen and 62 Ib. for helium, at
low altitude. Thus the envelope of a
balloon weighing 1 ton would, in the
taut state at sea-level, have a diameter
of 39-8 ft. for hydrogen and 41-1 ft.
for helium ; actually it would be
made larger, filling only at altitude
and being limp at sea-level.
FlG. 6. Referring to Fig. 6, OA represents

A.r>. — 1*

AERODYNAMICS

[CH.

the variation of atmospheric pressure from the level of the top of
the open filling sleeve S to that of the crest of the balloon, OH the
corresponding variation of pressure through the bulk of helium
filling the envelope. The difference between these external arid
internal pressures acts radially outward on the fabric as shown to
the right. The upward resultant force and part of the force of
expansion are supported by the net N, from which is suspended the
basket or gondola B, carrying ballast and the useful load.

Balloons drift with the wind and cannot be steered horizontally.
Airships, on the other hand, can maintain relative horizontal velocities
by means of engines and airscrews, and are shaped to streamline
form for economy of power. Three classes may be distinguished.

The small non-rigid airship, or dirigible balloon (Fig. l(a)} has a
faired envelope whose shape is conserved by excess gas pressure
maintained by internal ballonets which can be inflated by an air
scoop exposed behind the airscrew. Some stiffening is necessary,
especially at the nose, which tends to blow in at speed. A gondola,
carrying the power unit, fuel, and other loads, is suspended on cables
from hand-shaped strengthening patches on the envelope. (Only
a few of the wires are shown in the sketch.)

In the semi-rigid type (b) some form of keel is interposed between
the envelope and gondola, or gondolas, enabling excess gas pressure
to be minimised. Several internal staying systems spread the load
carried by the girder over the envelope, the section of which is not as
a rule circular.

The modern rigid airship (c) owes its external form entirely to a
structural framework covered with fabric. Numerous transverse
frames, binding together a skeleton of longitudinals or 'stringers,
divide the great length of the hull into cells, each of which accommo-
dates a gas-bag, which may be limp. Single gas-bags greatly
exceed balloons in size, and are secured to the structure by nets.
Some particulars of recent airships are given in Table I.

TABLE I

Airship

Zeppelins
Graf Hindenburg

R101

Akron (U.S.A.)

Length (ft.) .
Max. diam. (ft.) .
Gas used
Volume (cu. ft.) .
Approximate gross
lift (tons) .

776
100
Hydrogen
3-7 x 10«

112

800
135
Hydrogen
6-7 X 10*

203

777
132
Hydrogen
6-6 X 10a

167

785
133
Helium
6-5 X 10«

180

The largest single gas-bag in the above has a lift of 25 tons.

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 11

10. Centre of Pressure

Th£ point on a surface exposed to pressure through which the
resultant force acts is called the centre of pressure. The centres of
pressure with which we are concerned relate to the pressure differ-
ence, often called the gas pressure, unevenly spread over part of an
envelope separating gas from the atmosphere. Gas pressures are
small at the bottom of an envelope and reach a maximum at the top,
as illustrated in Fig. 6, and positions of the centres of pressure are
usually high.

The high centres of the total gas pressures exerted on walls which
restrain a gas-bag, as in the case of the wire bulkheads or transverse
frames of a rigid airship, lead to moments internal to the structure.

BCDE (Fig. 8) is a (full) gas-bag of an airship which is pitched at
angle a from a level keel. The longitudinal thrusts P, P' from the

* gas pressure ' are supported by bulkheads EC and DE of areas A,
A', assumed plane, B and E being lowest and C and D highest points.
The gas is assumed to be at rest, so that pressure is constant over
horizontal planes, and its pressure at B, the bottom of the bag, is
taken as equal to that of the atmosphere. Let p be the excess
pressure at height h above the level of B. Then from (3) p = pigA,
where px is the difference in the densities of the gas and the surround-
ing air.

Lower Bulkhead BC. — Let 8A be the area of a narrow horizontal
strip of BC distant y from a horizontal axis in its plane through B.
Then h = y cos a, and the total thrust on BC is given by :

re re

P = p dA = pig cos a y dA

JB JB

= P!# cos a . AyQ . . (i)

12 AERODYNAMICS [CH.

where y0 is the distance of the centroid of BC from the axis
through B.

Let the centre of pressure of P be distant y0 + Ay from the B-
axis, and take moments about this axis.

P(yQ + Ay) = py dA = ^g cos a y* dA
JB JB

= ?£ cos a . 7B . (ii)

where 7B is the second moment of the area about the B-axis. If 70
is this moment about a parallel axis through the centroid,
/B = /0 -f Ay <?. Substituting in (ii) :

__ Plg cos a (J0 + Ayl) _
AV-" p - y*

Hence from (i) :

where &0 is the radius of gyration.

The result is independent of pitch. For a circular bulkhead of
radius rf for example, /„ = nr*/4: and Ay = r/4. In practice, how-
ever, an excess pressure is often introduced, so that pB is not zero,
when a correction must be made, as will be clear from the following :
Upper Bulkhead DE.— Measuring now y in the plane of ED from
a parallel horizontal axis through E, we have :

P = 9iS(y cos a + / sin a),
where / is the distance apart of the bulkheads.

P' = Plg (y cos a + / sin &)dA
JE

— pig^'^o cos a + / sin a).

P'(yi + A/) = pig (y* cos a + yl sin <x.)dA

J E

= Ptf[(Ji + AW*) cos oi + A'yil sin a].
This gives

A ' - • .«i _ «'

^ ~ X + / tan a -X"

__ __^? M

""yj + /tana i;

The additional term in the denominator is EF. Hence (6) is
generalised by (7), since it is always possible to draw a horizontal line
BF at which any super pressure would vanish.

AIR AT REST, THE ATMOSPHERE AND STATIC LIFT

ii. Relation between Pressure, Density, and Temperature of a Gas

By the experimental laws of Boyle and Charles, for constant
temperature the pressure of a gas is proportional to its density ; for
constant volume the pressure of a gas is proportional to its absolute
temperature. The absolute temperature is denoted by T and, if 6 is
the temperature on the centigrade scale, is given by

T = 0 + 273.

Combining these laws, we have, for a given mass of a particular
gas:

pV^Bt (8)

where V is the volume, or, if V is the volume of 1 lb.,

P/9=gBi: (9)

B is a constant which is made characteristic of a particular gas by
treating 1 lb. of the gas ; it is then evaluated from measurements of
pressure and volume at a known temperature. It follows that B
will vary from one gas to another in inverse proportion to the
density under standard conditions of pressure and temperature.

If N is the ixumber of molecules in V, N will, by Avogadro's law,
be the same for all gases at constant p and T. Hence, writing pV/N
= B'T, B' is an absolute constant having the same value for all
gases. Equation (9) is more convenient, however, and the variation
of B is at once determined from a table of molecular weights.
Some useful data are given in Table II. It will be noticed that, if p
is kept constant, B measures the work done by the volume of gas in
expanding in consequence of being heated through unit temperature
change. The units of B are thus ft.-lb. per lb, per degree centigrade,
or ft. per ° C.

TABLE II

At 0° C.

and 760 mm.

mercury

Fluid

ft /°C

Ib./cu. ft.

cu. ft./lb.

P
slugs/cu. ft.

Dry air ...

0-0807

12-30

0-00261

06-0

Hydrogen
Helium

0-00561
0-01113

178-3
89-8

0-000174
0-000346

1381
606

Water-vapour

0-0501

10-05

0-00156

155

12. Isothermal Atmosphere

We now examine the static equilibrium of a bulk of gas under
gravity, taking into account its compressibility. Equation (2)

14 AERODYNAMICS [CH.

applies, but specification is needed of the relationship between p
and p. The simple assumption made in the present article is that
appropriate to Boyle's law, viz. constant temperature TO, so that />/p
remains constant. From (2) :

*.-*.

9g
From (9) : 1 _ J5r0

9g P '
Hence : ,.

BiQ = - dh.

Integrating between levels Ax and h2, where p = pt and p2 respec-
tively,

BTO log (pjpj = h, - h, . . . (10)

The logarithm in this expression is to base e. Throughout this
book Napierian logarithms will be intended, unless it is stated
otherwise. The result (10) states that the pressure and therefore
the density of a bulk of gas which is everywhere at the same tem-
perature vary exponentially with altitude.

The result, although accurately true only for a single gas, applies
with negligible error to a mass of air under isothermal conditions,
provided great altitude changes are excluded. The stratosphere is
in conductive equilibrium, the uniform temperature being about
— - 55° C. The constitution of the air at its lowest levels is as given
in Article 1. As altitude increases, the constitution is subject to
Dalton's law : a mixture of gases in isothermal equilibrium may be
regarded as the aggregate of a number of atmospheres, one for each
constituent gas, the law of density variation in each atmosphere
being the same as if it constituted the whole. Hence argon and
other heavy gases and subsequently oxygen, nitrogen, and neon will
become rarer at higher levels. The value of B for the atmosphere
will consequently increase with altitude, although we have assumed
it constant in order to obtain (10). The variation of B for several
miles into the stratosphere will, however, be small. At greater
altitudes still the temperature increases again.

13. The Troposphere

The atmosphere beneath the stratified region is perpetually in
process of being mechanically mixed by wind and storm. When a
bulk of air is displaced vertically, its temperature, unlike its pressure,
has insufficient time for adjustment to the conditions obtaining at the

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 15

new level before it is moved away again. The properties of this
part of the atmosphere, to which most regular flying so far has been
restricted, are subject to considerable variations with time and place,
excepting that B varies only slightly, depending upon the humidity.
There exists a temperature gradient with respect to altitude, and on
the average this is linear, until the merge into the stratosphere is
approached. It will be found in consequence that the pressure and
density at different levels obey the law —

P/9* = k ..... (11)

where k and n are constants. This relationship we begin by
assuming.

Substituting for p from (11) in (2) leads to

nk» »=*

----- m p n = __ gh _j__ const.
w ~— 1

Putting p = p0 when h = 0 gives for the constant of integration —

i
nk" «-i

Therefore :

M-l n-l n — I

To evaluate k let p0, TO, be the density and absolute temperature
atA=0. By (11):

while by (9) :

Hence :

PS \ P.
and

£ .... /rrT?^ \*A l-»

Substituting in (12)

p ( n — l h \£i

. (13)

n tQ v '

16 AERODYNAMICS [CH.

The temperature gradient is found as follows. From (9) and
(11):

n-l

=(!)"

Substituting in (13)

— 1 A

6 denoting temperature in ° C. This shows that while n remains
constant :

dt dQ n — 1

i.e. the temperature variation is linear.

As the stratosphere is approached, the law changes, the gradient
becoming less and less steep.

14. The International Standard Atmosphere

It is necessary to correct observations of the performance of air-
craft for casual atmospheric variation, and for this purpose the
device of a standard atmosphere is introduced. A number of
countries have agreed upon the adoption of an international standard,
representing average conditions in Western Europe. This is defined
by the temperature-altitude relationship :

6 = 15 — 0-00198116& . . . (17)

A being in feet above sea-level (the number of significant figures
given is due to h being expressed in the metric system in the original
definition). The dry air value of B, viz. 96-0, is also assumed.

This definition leads to the following approximations :
From (16) n == 1-235

From (13) p/p, = (1 - 0-00000688A)*'M'

From (14) pfp9 = (T/288)B-*" f • • I18)

Similarly P/PO == (T/288)4-a85

Some numerical results are given in Table III.

I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT

TABLE III

h (ft.)

0 (° C.)

PIP,

<r = p/Po

16-0

1-000

5,000

5-1

0-832

0-862

10,000

— 4-8

0-688

0-738

15,000

- 14-7

0-664

0-629

20,000

- 24-6

0-459

0-534

25,000

- 34-6

0-371

0-448

30,000

- 44-4

0-297

0-375

35,000

- 54-3

0-235

0-310

40,000

- 55

0-185

0-244

45,000

- 56

0-145

0-192

60,000

- 55

0-116

0-161

15- Application to Altimeters

A light adaptation of the aneroid barometer is used on aircraft,
with the help of a thermometer, to gauge altitude. To graduate the
instrument, increasing pressure differences are applied to it, and the
dial is marked in intervals of h according either to the isothermal or
to the standard atmosphere laws.

In the former case, the uniform temperature requiring to be
assumed is usually taken as 10° C. From (10) and Table III the
altitudes indicated are then excessive, on the basis of the standard
atmosphere, by about 1-6, 5-7, and 10 per cent, at altitudes of
10,000, 20,000 and 30,000 ft. respectively. Correction for decrease
of temperature with increase of altitude is made by assigning esti-
mated mean temperatures to successive intervals of altitude. Thus,
if TM applies to the true increase of altitude A#, corresponding to a
decrease of p from pl to p2, while AA is indicated by the altimeter
whose calibration temperature is TO, we have from (10)

A# = AA.^ . . . . (19)

Readings of altimeters with a standard atmosphere scale require
correction for casual variation of temperature. Let H, TH, and p
denote the true altitude, temperature, and pressure respectively,
and h the altimeter reading corresponding to p and the graduation
temperature T. Use suffix 0 for sea-level and write 5 for the tem-
perature lapse rate, so that T — TO == — sh. Then from (14), if s
remains constant —

t - 1

(i)

18 AERODYNAMICS [CH.

Hence :

To
giving

by (i).

. . . (20)

TO — Sfl v

1 6. Gas-bag Lift in General

The assumption of constant density made in Article 8 to obtain
expression (4) for the lift L of a gas-filled envelope may now be
examined. Although a balloon of twice the size has been con-
structed, 100 ft. may be taken as a usual height of large gas-envelopes.
The maximum variation from the mean of the air density then
follows from the formulae (18). At sea-level, where it is greatest, it
amounts to 0-15 per cent, approximately. Similarly, the maximum
variation of the air pressure from the mean is found to be less than
0*2 per cent. Equation (3) shows that the corresponding variations
in the gas will be smaller still.

Although the buoyancy depends on differences between atmos-
pheric and gas pressures, these are negligible compared with varia-
tions caused in both by considerable changes in altitude. Gas-bags
should be only partly filled at sea-level, so that the gas can, on
ascent, expand to fill an increased volume without loss.

To study the condition of a constant weight W' of gas enclosed,
(4) is conveniently written :

L = W'(±-\Y . . . (21)

We also have from (9), always distinguishing the gas by accented
symbols :

1 __ p __£V
a' ~ p ' ~~ ~B^
at all pressures, and, therefore, altitudes. So (21) becomes

• • • (22)

and it is seen that since B, B' are constants, L remains constant in
respect of change of altitude, provided that no gas is lost and that
no temperature difference arises between the gas and the surrounding
air. The last requirement involves very slow ascent or descent to

l] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 19

allow sufficient transference of heat through the envelope, or the
envelope must be held at a new altitude — as is possible by aero-
dynamic means with airships — until such transference has taken
place.

Gas-bags are too weak to support a considerable pressure, and
safety valves operate when they become full, leading to a loss of gas.
Thus the volume held in reserve at sea-level decides the maximum
altitude permissible without loss of gas. This is called the static
ceiling. A lighter-than-air craft can be forced to still greater alti-
tudes by the following means : aerodynamic lift ; heating of the gas
by the sun ; entering a cold atmospheric region ; or by discharging
ballast. The condition then is that V' remains constant. Exclud-
ing the case of variation of weight, we find from (4) that the gas lift
will remain constant only if p — p' remain constant or, by (9) if

P \ 5 W' )' i-e- if PI1 — HT7 ) remain constant. Hence, from

\ £>T D T / \ JD T /

(11) the condition is that n — _ Ljmust vary inversely as p*.

In this way it is simple to calculate the excess gas temperature
required for static equilibrium at a given altitude in excess of the
static ceiling. Gas having been lost, when the temperature differ-
ence vanishes ballast must be released for static equilibrium to occur
at any altitude.

17. Vertical Stability

The foregoing conditions depend upon the absence of a propulsive
or dragging force ; the envelope must move with the wind, otherwise
a variation of external pressure, different from that investigated,
may contribute to lift. A difference between gas-lift and total
weight, brought about by release of gas, for instance, or discharge
of ballast, creates vertical acceleration which leads to vertical
velocity relative to the surrounding air, equilibrium again being
attained by the supervention of an aerodynamic force due to the
relative motion. Variation of weight carried or of gas provides con-
trol of altitude, but even if, as in the case of airships, vertical control
by aerodynamic means is also possible, the practical feasibility of
lighter-than-air craft requires further investigation, since their level
of riding is not obviously fixed, as is the case with ships only partly
immersed in water. The first question is whether, in a stationary
atmosphere, a balloon would hunt upwards and downwards, restrict-
ing time in the air through rate of loss of gas due to the need for con-

20 AERODYNAMICS [ClL

tinual control. A second question is whether the atmosphere is
liable to continual up and down currents. These would have the
same effect on the duration of flight of a balloon, but the second
question has a wider significance, since such currents, if sufficiently
violent, would make flight by heavier-than-air craft also impossible.
Consider the rapid ascent of an envelope without loss of gas from
altitude hlf where the atmospheric pressure p = pt and the absolute
temperature T = TJ, to h2f where p = pt. When the atmosphere is
in standard condition we have :

/A\*~T /* \°t1903
T* = #A = (*!\

TX W \pj '

For the gas within the envelope the thermal conductivity is so small
that heat transference can be neglected. The gas then expands
according to the adiabatic law :

r = const.

Distinguishing properties by accented symbols, we have, since
Y= 1-405:

/ r,/x 0-288

T, (P*\

3 = (p'J •

Now assume that initially r( = TJ. Very closely, pl = p( and
pz = P*- Since p* < plf we then have that -ri < T2. Hence, by
Article 16 the gas-bag will sink, the load attached to it being con-
stant. Conversely, a rapid descent of a gas-bag results in temporary
excessive buoyancy.

Thus a lighter-than-air craft riding below its static ceiling tends to
return to its original altitude if displaced, provided displacement is
sufficiently rapid for passage of heat through the envelope to be
small. It is said to be stable in respect of vertical disturbance.
The state of the atmosphere is part and parcel of the question, for a
necessary proviso is seen to be that n < y.

If the craft is above its static ceiling, the stability in face of down-
ward disturbance is the same, since no further gas is lost. But for
upward displacement the stability is greater, since the weight of gas
enclosed decreases.

1 8. Atmospheric Stability and Potential Temperature

The foregoing reasoning may be applied to the rapid vertical dis-
placement of a bulk of the atmosphere, and we find that if, for the

l] AIR AT REST, THE ATMOSPHERE AND STATIC IIFT 21

atmosphere, n has a value less than 1-405, up and down currents are
damped out. If n = 1-405, the stability is neutral ; the atmosphere
is then said to be in convective equilibrium. When n > y, a condi-
tion that may arise from local or temporary causes, vertical winds
occur and make aeronautics dangerous, if not impossible.

The condition for atmospheric stability is discussed alternatively
in terms of ' potential temperature/ The potential temperature at
a given altitude is defined as the temperature which a given bulk of
air at that altitude would attain if displaced to a standard altitude,
such as sea-level, the compression taking place without loss or gain
of heat. We see that f or n = y the potential temperature would be
the same for all altitudes. For n < y the potential temperature
increases upward. An atmosphere is stable when the potential tem-
perature is greater the greater the altitude. It will be noticed that
the stratosphere is more stable than the troposphere.

19. Bulk Elasticity

Fluids at rest possess elasticity in respect of change of volume.
The modulus of elasticity E is defined as the ratio of the stress caus-
ing a volumetric strain to the strain produced. An increase of
pressure from ptop + $p will change a volume V to V — W. The
strain is — 8V I V and E is given by

'-*/(-?)•

Since

£ = P ..... (23)

tfp

In the case of liquids the compressibility is very small, and (23)
sufficiently defines Et but with gases we must specify the thermal
conditions under which the compression is supposed to take place.
The interest of E in Aerodynamics is chiefly in respect of changes of
pressure, and therefore of density, occurring in air moving at approxi-
mately constant altitude. The changes are usually too rapid for
appreciable heat to be lost or gained, having regard to the small
thermal conductivity of air. In these typical circumstances the
adiabatic law is again assumed, viz. p = &pv, so that from (23)

E =

22 AERODYNAMICS [CH. I

20. Velocity of Sound

The condition under which (24) has been derived is ideally realised
in the longitudinal contractions and expansions produced in elements
of the air by the passage of waves of sound. Newton demon-
strated the following law for the velocity a of such waves in a homo-
geneous fluid :

a = V(£/p).

Thus for gases, from (24) —

a = Vy^/p .... (25)
or, substituting from (9),

a = VjgBi: (26)

The velocity of sound is seen to depend on the nature of the gas and
its temperature only.

We shall always employ the symbol a for the velocity of sound in
air. With y = 1-405, g = 32-173 and B = 96-0,

a = 65-9-v/r .... (27)
nearly. For 15° C., T = 288,

a = 1118 ft. per sec. . . . (28)

A disturbing force or pressure suddenly applied to a part of a solid
body is transmitted through it almost instantaneously. From the
preceding article we infer that through air such a disturbance is pro-
pagated more slowly, but yet at a considerably greater rate than the
velocities common in aeronautics. Disturbance of the stationary
equilibrium of a bulk of air follows from swift but not instantaneous
propagation through it of pressure changes. It may be noted, for
example, that a moving airship disturbs the air far in front of it ; a
fast bullet, on the other hand, overtakes its propagation of distur-
bance and fails to do so. This change assumes great significance
in connection with stratospheric flying, for two reasons : a decreases
to between 970 and 975 ft. per sec., the flight speeds of low altitude
are at least doubled to compensate for the reduced density of the
air.

Chapter II
AIR FLOW AND AERODYNAMIC FORCE

21. Streamlines and Types of Flow

It is familiar that motions of air vary considerably in character.
Means of discriminating with effect between one kind of flow and
another will appear as the subject develops, but some preliminary
classification is desirable.

Streamlines. — Discussion is facilitated by the conception of the
streamline. A streamline is a line drawn in the moving fluid such
that the flow across it is everywhere zero at the instant considered.

Uniform Flow. — The simplest form of flow is uniform motion. By
this we mean that the velocity of all elements is the same in magni-
tude and direction. It follows that the streamlines are all parallel
straight lines, although this is not sufficient in itself to distinguish
uniform motion.

Laminar Parallel Flow. — There are other motions whose stream-
lines are parallel straight lines, In which the velocity of the element,
although uniform in direction, depends upon distance from some
fixed parallel axis or plane. Such motions are properly called
laminar, although the name laminar is nowadays frequently used in a
wider sense, strictly laminar motions being characterised as ' parallel/

Both uniform and laminar motions are steady, i.e. the velocity at
any chosen position in the field of flow does not vary in magnitude or
direction with time. They are more than this, however, for the
velocity of any chosen element of fluid does not vary with time as it
proceeds along its path. (It is specifically in this respect that wider
use is commonly made of the name laminar.)

General Steady Flow. — We may have a steady motion which is
neither uniform nor strictly laminar. The streamlines then form a
picture of the flow which does not vary with time, but the velocity
along a streamline varies from one position to another. Thus the
elements of fluid have accelerations. The streamlines are riot
parallel and in general are not straight. It is this more general
kind of flow that is usually intended by the term ' steady motion '
used without qualification.

24 AERODYNAMICS [Cfl.

Unsteadiness and Path-lines. — Steady motions are often called
' streamline/ All steady motions have one feature in common :
the streamlines coincide with the paths of elements, called path-
lines.

Unsteady motions are common in Aerodynamics, and in these the
path-lines and streamlines are not the same. The velocity varies
with both space and time. At a chosen instant streamlines may be
drawn, but each streamline changes in shape before an element has
time to move more than a short distance along it. An unsteady
motion may be such that an instantaneous picture of streamlines
recurs at equal intervals of time ; it is then said to be periodic or
eddying, though use of the latter term is less restricted.

Turbulence. — When unsteadiness of any kind prevails, the motion
is often called turbulent. In addition to periodic we may have
irregular fluctuations. These may occur on such a scale that
transient streamlines might conceivably be determined. But in
other cases the fluctuations are much more finely grained, conveying
the impression of a chaotic intermingling of very small masses of the
fluid accompanied by modifications of momentum. This last type
of unsteadiness is, unfortunately, at once the most difficult to under-
stand and the most important in practical Aerodynamics. It has
come to be the form usually intended by the name turbulence.

Stream-tube. — A conception of occasional use in discussing steady
flow is the stream-tube. This may be defined as an imaginary tube
drawn in the fluid, of small but not necessarily constant section,
whose walls are formed of streamlines. Clearly, no fluid can enter
or leave the tube through the walls except in respect of molecular
agitation.

Two-dimensional Flow. — Another conception, of which we shall
make very frequent use, is that of two-dimensional flow. Consider
fixed co-ordinate axes Ox, Oy, Oz drawn mutually at right angles in
the fluid. Let the velocity components of any element in the direc-
tions of these axes be u, vt and w9 respectively. Two of the direc-
tions, say Ox and Oy, are open to selection, the third then following.
If the motion is such that we can select Ox, Oy in such a way that
w = 0 for all elements at all times, and also if neither u nor v then
vanishes, the motion is of general two-dimensional form. The
streamlines drawn in all planes parallel to a selected #y-plane will be
the same. It is then sufficient to study the motion in the ^y-plane,
tacitly assuming that we are dealing with a slice of the fluid in
motion of unit thickness perpendicular to this plane.

If besides w = 0 we have another velocity component, say vt

II] AIR FLOW AND AERODYNAMIC FORCE 25

everywhere vanishing, the motion is strictly laminar, or parallel,
and we may have u depending either upon distance from the plane
xOz or upon distance from the axis Ox. In the former case, where u
is a function of y only, the motion is two-dimensional ; in the latter,
the flow is of the kind that occurs in certain circumstances along
straight pipes of uniform section, when it is sufficient to consider unit
length of the pipe because the distribution of flow will be the same
through all cross-sections.

22. Absence of Slip at a Boundary

The theorem of Article 3 holds equally for a fluid in uniform
motion if the rigid surface exposed in the fluid moves exactly with
it. The pressure in uniform motion is thus constant and equal in all
directions. Unless the whole motion is uniform, however, the
theorem fails, and considerable investigation is necessary to establish
precisely what we then mean by ' pressure/

Imagine a small, rigid, and very thin material plate to be im-
mersed and held stationary in the midst of a bulk of air in motion ;
let its plane be parallel to the oncoming air, considered for simplicity
to be in uniform motion. The disturbance caused by the plate
might, on account of its extreme thinness, be expected to be neglig-
ible. This would, however, be completely at variance with experi-
mental fact. Experiment clearly shows that the fluid coming into
contact with the tangential surfaces of the plate is brought to rest,
whilst fluid that passes close by has its velocity substantially
reduced.

To explain this phenomenon in molecular terms we may suppose
the plate to be initially chemically clean, each surface being a
lattice-work of atoms of the substance of which the plate is made.
As such it exposes a close distribution of centres of adhesive force.
The force of adhesion is very intense at distances from the surface
comparable with the size of a molecule, and a molecule of gas
impinging on the surface is held there for a time. Considering the
whole lattice-work, we may say that the air is condensed on it, since
the molecules no longer possess a free path. But the layer of con-
densed gas receives energy, partly from the body of the plate and
partly from bombardment by free gas molecules, and where the
energy attains to the latent heat of evaporation the molecules free
themselves and return to the bulk of the gas— thereby only giving
place, however, to others. Thus the film of condensed gas molecules
is in circulation with the external free gas.

Regarding the action of the plate on the stream of air, we must

26 AERODYNAMICS [CH.

suppose, therefore, two effects to result from molecular constitu-
tion : (a) impinging air molecules are brought to rest relative to the
bulk or mass motion — just as they are, for a time, in regard to the
molecular motion ; (b) air molecules released from the plate are de-
prived of mass motion, and, requiring to be accelerated by the other
molecules, retard the general flow to an appreciable depth. Thus
the rate of change of molecular momentum at the plate is no longer
normal to its surface, but has a tangential component ; in other
words, the ' pressure ' on the plate is oblique. Further, the retarda-
tion occurring at some distance into the fluid shows that the pressure
in this affected region away from the plate cannot be equal in all
directions. It will be noted that the mass flow is no longer uniform ;
its initial uniformity has been destroyed by introducing the plate
which has a relative velocity.

The phenomenon of absence of slip at the surface of separation of a
material body from a surrounding fluid occurs quite generally and is
of fundamental importance in Aerodynamics. It is known as the
boundary condition for a real fluid. No matter how fast a fluid,
gaseous or liquid, is forced to rush through a pipe, for example, the
velocity at the wall is zero. The velocity of the air immediately
adjacent to the skin of an aeroplane at any instant is equal to that of
the aeroplane itself.

Thus a uniform fluid motion cannot persist in the presence of
a material boundary which is not moving with the same velocity
(although the motion may remain steady). The * pressure ' at a point
in the unevenly moving fluid will depend upon the direction con-
sidered. The matter is further investigated in the following articles.

VISCOSITY
23. Nature of Viscosity

If air is moving in other than uniform motion, a further physical
property is brought into play in consequence of the molecular
structure of the fluid. Its nature will be discussed with reference to
laminar (or parallel) two-dimensional flow. Let this flow be in the
direction Q%% and draw Oy so that u, the mass velocity, is a function
of y only.

Consider an imaginary plane, say y =y't perpendicular to Oy
(Fig, 9). This plane is formed of streamlines, but owing to mole-
cular motion, molecules are continually darting across it in all direc-
tions. Density remains uniformly distributed, and this condition
entails that the same number of molecules crosses a chosen area of

AIR FLOW AND AERODYNAMIC FORCE

the plane in unit time from either side. The molecules possess, in
addition to their molecular velocity, a superposed mass velocity ut
which by supposition is different on one side of the plane from on the
other. Hence molecules crossing in one direction carry away, per
unit area of the plane and in unit time, a different quantity of mass
momentum from that which those crossing in the opposite direction
bring with them. Hence momentum is being transported across the
direction of flow. This phenomenon is called the viscous effect.
Clearly, it exists only in the presence of a velocity gradient, which it
tends to destroy in course of time.

Qualitative Theory of Viscosity

Consider an imaginary right cylinder (Fig. 9) of unit length and
unit cross-section, whose ends are parallel to and, say, equidistant
from the imaginary plane
y = yf. Denote by 5 the
unit area of the plane which
the cylinder encloses. If p
is the density of the air and
m the mass of each molecule,
the number of molecules
within the cylindrical space
is p/w and is constant.
These molecules are moving
in all directions with a
mean molecular velocity c FIG. 9.

along straight paths of mean

length X. They have in addition a superposed mass velocity u
whose magnitude depends upon their values of y at the instant
considered, subject to the consideration that the u of any particular
molecule cannot be modified while it is in process of describing a free
path, for changes can come only from collisions.

The molecular velocities of all the p/w molecules can be resolved
at any instant parallel to Ox, Oy, Oz, but, as in Article 4, the number
being very large, this procedure may be replaced statistically by
imagining that p/6w molecules move at a velocity c in each of the
two directions which are parallel to each of the three co-ordinate
axes. This equivalent motion must be supposed to extend through
the interval of time At which is required for a displacement of the
molecules through a distance X. Thus A/ = X/c. At the end of this
interval collision occurs generally.

We are concerned only with molecules which cross S, and so ignore

28 AERODYNAMICS [CH.

all moving parallel to Ox, Oz. Of molecules moving parallel to Oyt
only those within a distance X of y = y' can cross during Atf. Thus,
S being unity and there being no displacement of mass, Xp/6w
molecules cross in each direction during this time. For clarity we
shall speak of y increasing as ' upward ' and assume u to increase
upward. There is also no loss of generality in supposing that all
molecules penetrating S from above or below y9 start at distance X
from that plane, the velocity at y =y' being u.

Consider a single exchange by the fluid above y'. It loses on ac-

/ du
count of the downward-moving molecule momentum =mlu + ^-

whilst it receives by the upward-moving molecule momentum = mu,

a loss on this account of wX— . But in addition it must, by

collision at the end of A£, add momentum to the incoming molecule

to the amount wX— . Thus the total change in the momentum of
Ay

the fluid above y' in respect of a single molecule exchanged with one

du
from below is a loss amounting to 2wXT-. Summing for all pairs, the

aggregate loss is :

- .
dy

The rate of this loss is :

1 pX2 du

AV 3 dy
or, since A£ = X/c, the rate is

The rate of change of mass momentum being parallel to Ox, it may
be represented by a force in the fluid at^ =y' acting tangentially on
the fluid above. If the intensity of this traction is F, we have, since

The direction of F is such as to oppose the motion of the fluid above.
Similarly, we find that the fluid below / gains momentum at the
;ame rate. We note the passage downward of momentum and that
i traction F acts at S in the opposite direction on the fluid below,
irging it forward.

II] AIR FLOW AND AERODYNAMIC FORCE

The coefficient by which du/dy is to be multiplied in order to
determine F is called the coefficient of viscosity, and is denoted by JA.
Its dimensions are (Af/Z8) . (LIT) . L = M/LT.

24. Maxwell's Definition of Viscosity

The following example is instructive from several points of view.
A number of layers of air, each of thickness ht are separated from one
another by a series of infinite horizontal plates. Alternate plates are
fixed, while the others are given a common velocity U in their own
planes. The resulting conditions in all layers will be the same
except fcfor a question of sign, and we shall investigate one layer
only.

Draw Ox (Fig. 10) in the fixed plate (taken to be the lower one)

u=U

U-O

FIG. 10.

and in the direction of motion of the other, and Oy vertically up-
ward. By Article 22 air touching the fixed plate has a velocity
u = 0, while for air touching the moving plate u = U, and the
fluid between is urged forward from above, but the ensuing motion
is retarded from below.

Now it is assumed that, after sufficient time has elapsed, the
motion in the layer becomes steady. In these circumstances con-
sider a stratum of air of thickness 8y between the plates and parallel
to them. If the velocity at the lower face distant y from Ox is u,

du
that on the upper face is u + ^~8y. The intensity of traction F on

the lower face is equal in magnitude to u,—~, and retards the stratum ;

that on the upper face is jx ~ ( u + -- 8y Y and tends to accelerate the

dy\ dy /

stratum. The resultant traction on the stratum in the direction Ox

. j d ( du
1S ^_ + _

du\ d*u ^
— j- r = pi j -8y. But as the motion is steady.

30 AERODYNAMICS [CH.

there cannot be a resultant force on the stratum. Hence :

Integrating twice,

u = Ay + B

where A and B are constants of integration. Now insert in this
equation for u the special values which are known, viz. u = 0 when
y = o, w = U when y = h. Two equations result, viz. :

0 =o + B

U =Ah + B

which are sufficient to determine A and B. We find :

5 =0

A = Z7/A.
Inserting these values in the original equation for u,

u = j,y ..... (30)

Thus the fluid velocity between the plates is proportional to y . The
distribution of velocity is plotted in the figure.

Let F be, as before, the intensity of traction, and reckon it positive
in the direction Ox. The traction exerted on the fluid adjacent to
the lower plate by the fluid above is given by

This traction is transmitted to the lower plate and a force of equal
intensity must be applied in the opposite direction to prevent it
from being dragged in the Ox direction. Similarly, it is found that a
force

F= ^

must be applied to the upper plate to maintain the motion. But
from (30) —

/du\ __ E7 _ /du

)y^ ~~ h ~~~ \

Hence the forces on the plates are equal and opposite, as is otherwise
obvious. F is, in fact, uniform throughout the fluid. Hence this
case of motion is known as uniform rate of shearing.

If U = 1 = A, the intensity of either force is equal to (z. Hence,
Maxwell's definition of the coefficient of viscosity as the tangential

U] AIR FLOW AND AERODYNAMIC FORCE 31

force per unit area on either of two parallel plates at unit distance
apart, the one being fixed while the other moves with unit velocity,
fMfiuid filling the space between them being in steady motion. ,
' In the general case the moving plate does work on the layer of
fluid at the rate fjtC7a/A per unit area of the plate. The result is a
gradual rise in temperature of the fluid unless the heat generated is
conducted away.

25. Laws of Viscosity

The traction on a bounding surface past which a fluid is flowing is
called the skin friction. It differs in nature from the rubbing friction
between two dry surfaces, but is essentially the same as the friction
of a lubricated surface, such as that of a shaft in a bearing.

In certain cases of laminar flow, as will be seen in Chapter IX, the
boundary value of the velocity gradient can be calculated in terms
of a total rate of flow which can be measured experimentally, while
the skin friction can also be measured. Hence the value of \i can be
deduced without reference to the theory of Article 23. By varying
the density, pressure, and temperature of the fluid in a series of
experiments, empirical laws expressing the variation of y. can be
built up.

The experimental value of y. for air at 0° C. is given by

N = 3-58 x 1(T7 slug/ft, sec. . . (31)

It is interesting to compare this with a numerical value obtainable
from the qualitative theory. Equations (1) and (9) of Chapter I
together give

c* = 3gJ3T ..... (32)

The value of c calculated from this expression for 0° C., viz. 1591
ft. per sec., is greater than the mean molecular velocity given in
Article 1 for this temperature, because it is a root-mean-square value.
Hence, according to (29) —

(i)

giving, for 0° C., pt = 2-58 x 1G~7. The error, amounting to 28 per
cent., is removed by more elaborate analysis, which shows that the
mean free path must effectively be increased in the viscosity formula.
But this mathematical development is not required in Aerodynamics,
since JJL can be measured accurately.

The first law of viscosity is that the value of the coefficient is
independent of density variation at constant temperature. This

AERODYNAMICS

[CH.

surprising law is expressed in (i), because it is obvious that X must be
inversely proportional, approximately, to p. After predicting the
law, Maxwell showed it experimentally to hold down to pressures of
0-02 atmosphere. It tends to fail at very high pressures.

According to (i) a second law would be [i oc VT, but experiment
shows [i to vary more rapidly with the temperature. An empirical
law for air is

• • ' • (33>

(Rayleigh), where (JLO is the value of the coefficient at 0° C.

PRESSURE IN AIR FLOW
26. Relation between Component Stresses

We now prove a relationship that exists between the stresses on
an element (in the sense of Article 2) of a fluid in any form of two-
dimensional motion. In the
general case we have four com-
ponent stresses to deal with,
and a certain nomenclature is
adopted, as follows. For a
face drawn perpendicular to
Ox, the normal pressure on it
in the direction Ox is denoted
by pxx and the tangential com-
ponent in the direction Oy by
pxr The corresponding normal
and tangential pressures on a
face perpendicular to Oy are

FIG. 11.

,and^.

It will always be possible to find two axes at right angles to
one another, moving with the element, such that, at the instant
considered, the pressures in these directions tend to produce either
simple compression or simple dilatation in the element. These axes
are called principal axes and the pressures in their directions principal
stresses.

Let G be the centre of the element which is moving in any manner
in the plane xOy. Let Gxf, Gy' be the principal axes at any instant,
inclined at some angle a to the fixed axes of reference Oxt Oy.
Denote by p^ (Fig. 1 1) the principal pressure parallel to Gx' and by p^
that parallel to Gy' , and take a negative sign to indicate that the

II] AIR FLOW AND AERODYNAMIC FORCE 33

pressure is tending to compress the element and a positive sign that
it is tending to dilate it.

Adjacent to G draw X'Y' perpendicular to Oy and X'Y " of equal
length perpendicular to Ox, forming with the principal axes the
element-triangles GY'X' and GX"Y" (Fig. 11). These triangles are
to be regarded as the cross-sections of prisms A and B, respectively,
which have the same motion as that of G and whose faces are per-
pendicular to the Ay-plane. Let X'Y' = X"Y" = A, then A is
equal to the area of each of these two particular faces of the prisms
per unit length perpendicular to the #jy-plane. Similarly the area per
unit length of the GX' face = that of the GY" face = A cos a, etc.

The prisms of fluid form part of the general motion and have
accelerations. The forces arising from these are proportional,
however, to mass, i.e. to A8, and, as A is supposed very small, are
negligible compared with forces arising from the stresses, which are
proportional to A2. Hence the stresses are related by the condi-
tion for static equilibrium.

For the equilibrium of prism A we have, first, resolving in the
direction Ox —

y
or

pyx .A — p1 . A sin a . cos a + P* • A cos a . sin a = 0,

Pyx = (Pi — ^2) sin a cos a.
;cti(

pxy . A — />! . A cos a . sin a + pz • A sin a . cos a = 0,

Resolving in the direction Oy, we have, in regard to the equilibrium
of B—

Pxy = (Pi — P*) sin a COS a«

Hence :

Ay = Py* = i(#i — #«) sin 2a. . . (34)

The pressure pxy is identical with the tractional stress F of equation
(29) and involves an equal tractional stress at right angles. This
conversely is the condition for principal axes to exist.

With regard again to the equilibrium of A, but resolving now in the
direction Oy —

pyy . A — pl . A sin2 a — p* . A cos1 a = 0,
while resolving parallel to Ox with regard to B —

pxx . A — p1 . A cos* a — pt . A sin2 a = 0.
These two equations together give :

A.D.— 2

34 AERODYNAMICS [CH.

This equation is independent of a. Hence the arithmetic mean of
the normal components of pressure on any pair of perpendicular
faces through G is the same.

27. The Static Pressure in a Flow
Let us write :

-#=*(#! + PI = *(#« + Pyy) • • (36)

where, it has been found possible to say, x and y are any directions at
right angles to one another. Then p does not depend upon direction
and is the compressive pressure we shall have in mind when referring
to the ' static pressure/ or simply the pressure, at a point of a fluid
in motion. (The system of signs adopted in the last article will be
found convenient in a later chapter.) It will be noted that, if the
fluid were devoid of viscosity, p would be the pressure acting equally
in all directions at a chosen point, although not necessarily equally
at all points.

The basis of the experimental measurement of p is as follows.
The mouth of the short arm of an L-shaped tube is sealed, and a ring
of small holes is drilled through the tube wall a certain distance from
the closed mouth. The long arm is connected to a pressure gauge,
so that the outer air communicates with the gauge through the ring
of holes. The other side of the gauge is open to the atmosphere.
The tube is then set in motion in the direction of its short arm
through approximately stationary air. It is apparent, from Article
22, that the pressure acting through the ring of holes will not in
general be the same as with the tube stationary. Nevertheless, a
design for the short arm can be arrived at by experiment, such that
the gauge shows no pressure difference when the tube is given any
velocity, large or small. Adding to the whole system of tube and air
a velocity equal and opposite to that of the tube converts the case of
motion to that of a stationary tube immersed in an initially uniform
air-stream. The pressure communicated is then the same when the
tube is immersed in uniform flow or in stationary air. Thus the tube
correctly transmits the static pressure of a uniform motion. To cope
with motions in which the static pressure varies from point to point,
the tube may be reduced to suitably small dimensions ; even 0«5 mm.
diameter is practicable, the ring of holes then degenerating to one or
two small perforations.

28. Forces on an Element of Moving Fluid

The forces on the three-dimensional element fcSyftr are con-
veniently grouped as due to (a) external causes, such as gravity,

nj AIK FLOW AND AERODYNAMIC FORCE 35

(b) variation of the static pressure p through the field of flow, (c)
tractions on the faces.

In regard to (a) it may be remarked generally that, although air-
craft traverse large changes of altitude, the air motions to which they
give rise are conveniently considered with the aircraft assumed at
constant altitude and generalised subsequently. The air will be
deflected upward or downward, but its changes of altitude are then
sufficiently small for variations of density or pressure on this account
to be neglected. An element of air may be regarded as in neutral
equilibrium so far as concerns the gravitational field, its weight being
supposed always exactly balanced by its buoyancy.

(b) We shall require very frequently to write down the force on an
element due to space variation of p. Choose Ox in the direction in
which p is varying and consider the forces due to p only on the
faces of the element 8x8y8z. The forces on all the 8x8y and 8x8z
faces cancel, because p is varying only in the ^-direction. On the
8y8z face that is nearer the origin, the force is p . 8y8zf while on that

farther from the origin the force is (p + ~ 8x j 8y8z.
The resultant force in the direction Ox is thus

p . 8y8z -(P + '-r- 8*)

= — -~- X the volume of the element . . (37)
dx * '

This result should be remembered.

(c) The tractions have already been discussed to some extent.
They are proportional to p,, which is small for air. Close to the sur-
faces of wings and other bodies studied in Aerodynamics, the velocity
gradients are steep and the tractions large. Away from these
boundaries, however, the velocity gradients are usually sufficiently
small for the modification of the motion of the element due to the
tractions to be neglected.

BERNOULLI'S EQUATION
29. Derivation of Bernoulli's Equation

The following five articles treat of flow away from the vicinity of
material boundaries, and such that the tractions on the element can

36 AERODYNAMICS [CH.

be neglected, i.e. the pressure p is assumed to act equally in all direc-
tions at any point. It is also assumed that the flow is steady.
Consider steady flow of air at velocity q within a stream-tube

(Article 21) of cross-sectional area A
(Fig. 12). Denote by s distance
measured along the curved axis of
the tube in the direction of flow.
The condition of steady motion
means that p, q, p, A may vary
with s, but not, at any chosen posi-
FlG 12> tion, with time. Since fluid does not

collect anywhere —
pqA = constant . . . . (38)

The volume of a small element 8s of the air filling the stream-tube
is A 8s, and the force on it in the direction of flow due to the pressure

dp
variation is — - . ,48s, by (37). The mass of the element is

and its acceleration is dq/di. By Newton's second law of motion—

But

Jt = ds ' It = qds*
Hence :

I dp , dq

~~~ + q-~^=Q (39)

p ds ds v '

Integrating along the stream-tube, which may now be regarded as
a streamline —

h l?a = constant. . . . (40)

J p v '

This is the important equation of Bernoulli. Evaluation of the
remaining integral requires a knowledge of the relationship between
p and p. The constant appertains, unless proved otherwise, only to
the particular streamline chosen ; it must be regarded in general as
varying from one streamline to another of the same flow. Another
form is obtained by integrating (39) between any two values of s,
where the conditions are denoted by 1 and 2 :

{41)

II] AIR FLOW AND AERODYNAMIC FORCE 37

30. Variation of Density and Temperature

1°. — Let us first assume the flow to be isothermal, so that p and p
obey Boyle's law : dp /dp = constant. The integral remaining in
(40) may then be written :

dp _

* P Y P

from (25), a being the velocity of sound and y the ratio of the
specific heats = 1-405. This reduction is possible because (Article
20) a remains constant under the isothermal condition. Hence (41)
becomes, on evaluating the integral —

J <

T Pi
or

Pi
Expanding in an exponential series —

Pl «

under isothermal conditions.

2°. — Density variations actually occur so rapidly in, most aero-
dynamical motions that the isothermal assumption is inappropriate,
and, in fact, the condition is closely approached that no heat is lost
or gained. The adiabatic law then relates the pressure to the den-
sity, viz. :

p = kf . . . . (44)

The absolute temperature T now varies from point to point according
to

. (45)

From (44) :

dp — Y*pY - l dp.
Thus :

f*dp (• Y

— = Y* P

J i P J i

38 AERODYNAMICS

or, eliminating k by (44),

1 pi l\pl>

'-l} • . . (46)

where, it will be noted, the velocity of sound introduced from (25)
refers to the position slt where T = Tt.
Substitution in (41) leads to—

— 1) — ~- > . . (47)

Finally, expanding by the Binomial Theorem—

— = 1 2 — i- J- -_ I[±l iL | __ (A0\

pi 2dj* \2_ \ 2a^ / * '

Comparison with (42) and (43) shows density variation now to be
less, also that the convenient expression —

P~ = * Ul • . . (49)

applies closely to adiabatic flow, provided the velocity change is not
great. If qj — q* amounts to \a? the error in (49) is only 1-3 per
cent. ; this would occur, for example, if <72 = 2^ = 912 ft. per sec.,
or if q* = 3#x = 838 ft. per sec.

There is an important limit to the application of (47) ; q2 cannot
exceed a2, because #a gives the limiting velocity with which pressure
waves can be propagated. It will be noted that, since the tempera-
ture is reduced on expansion, 0a < ax. When q2 = a2 and ql = 0,
we find the minimum value of the density ratio :

But

/-i

Hence :

Minimum ?? = (-^y "... (50)

If TJ = 288, i.e. Oj = 15° C., this gives 0-634 and £2 max. = a9 =

II] AIR FLOW AND AERODYNAMIC FORCE 39

1019 ft. per sec. The final temperature is — 33*7° C., a drop of
48-7° C.

The examples worked out in Table IV further illustrate adiabatic
flow. Two cases of common occurrence are studied : (a) a stream
brought to rest (q^ = 0), (b) the velocity doubled (qn = 2^). In all
cases the initial conditions assumed are : pl = 760 mm. mercury,
Oj = 15° C. The values of Ai\A± are obtained from (38), by which,
since q, = 2ql9 AJA^ = J(p!/p»).

TABLE IV
EXAMPLES OF ADIABATIC FLOW

ft (ft. per sec.)

100

200

300

400

621

— ™ (per cent.)

0-4

1-8

3-6

6-5

11-1

M°C.) .

15-4

16-8

19-2

22-5

27-7

2?,

Pi - Pi / ccnt %

- 1-2

-4-7

- 10-5

- 18-2

-30

Pi v±~* """"'

2?,

<?a (° C.) .

13-6

9-4

2-4

- 7-4

- 23-1

2ft

^^t • • •

0-50(5

0-525

0-56

0-61

0-71

The variation of temperature affects such questions as the trouble-
some formation of ice on wings and the location of convective
radiators. Otherwise it is ignored.

31. Variation of Pressure — Comparison with Incompressible Flow

Equations for pressures corresponding to those of the preceding
article for densities are obtained in a similar way. They follow
immediately, however, by use of the relations : pi/p* = pi/pt for
isothermal and pi/p* = (pi/p2)Y for adiabatic flow. Thus Ber-
noulli's equation for adiabatic flow, which alone will now be con-
sidered, is found, with the help of (46), to be —

•r-}

Y-l
This gives, corresponding to (47) —

= 0.

• (51)

(52)

Now an outstanding result of the investigation of density variation
is that it is small provided velocities do not approach that of sound.

40 AERODYNAMICS [CH.

The condition p = constant is then a first approximation. Making
this assumption gives at once, from (40) —

P + ip?a = constant . . . (53)

for incompressible flow along a particular streamline, provided
always that tractions can be neglected. (41) becomes —

of which convenient non-dimensional forms are

. . . (55)

^=A'-*UrV-i|- • • • <56)

These alternative expressions of Bernoulli's theorem for an incom-
pressible fluid are of great importance.

We now determine the error involved in applying (53) to a gas
which is flowing adiabatically. Expanding (52) by the binomial
theorem —

fr = 1 ~ 2 ~~^7~~ + sv «7/ ""••••

Since Y^>I/«IS == p! by (25), this reduces, with r written for q^qit to

A similar expression is readily obtained to compare with (56).

The above series is rapidly convergent, and the equation indicates
that the error involved in applying (55) to a gas in adiabatic flow is
small, provided that q? is small compared with a,2. Since a± is only
approached by ql in particular cases, as for example at the tips of
airscrews, it follows that air in motion may usually be treated as an
incompressible fluid, such as water.

As an example, consider the case q2 = 2qt. The error involved
in employing (55) instead of (57) is as follows :

ql (ft. per sec.) : 100 200 300 400

y, (ft. per sec.) : 200 400 600 800

error (per cent.) : 0-6 2-4 5-5 10

II] AIR FLOW AND AERODYNAMIC FORCE 41

32. The Pitot Tube and the Stagnation Point

Consider an L-shaped tube immersed and held stationary in a
stream, one arm being parallel to it with open mouth direetly facing
the oncoming air ; suppose the other end to be connected to a pres-
sure gauge so that no air can flow through. There must exist an
axial streamline about which fluid approaching the mouth divides in
order to flow past. Air following that streamline will arrive at some
point within the mouth of the tube, where the time-average of the
velocity is zero ; we refrain from saying that the velocity will be zero,
because some unsteadiness may possibly exist in the mouth of the
tube ; but we can assert that the time-average of the square of the
velocity will be negligible in ordinary circumstances, compared with
the square of the velocity of the oncoming stream. Denote by p,
q, p, a the pressure, velocity, density, and the velocity of sound at a
point of the streamline far upstream, and use suffix 0 for the mouth
of the tube. Ignoring the small unsteadiness that may arise, and
also, for the moment, variation of density, the pressure p0 in the
mouth of the tube is given from Article 31 by :

.... (58)

Such a tube is called a pitot tube (after its eighteenth-century
inventor), and p0 the pitot head, or total head, for air flow whose
changes of pressure due to variation of altitude can be neglected.
Comparing with (53), we note that the constant of that equation is
measured by a pitot tube. Variation of p0 from one streamline to
another is readily determined by a pitot tube in experiment, its
diameter being made very small where the space-variation of total
head is rapid. For accurate work the tube must be oriented to lie
parallel to the local streamlines of the flow.

Variations of pQ are small compared with^>, and it is convenient to
deal with the quantity p0 — p, sometimes called the dynamic head.
For incompressible flow, to which Bernoulli's equation applies, we
have pQ — p = Jpy2. For the corresponding flow of a gas we find,
in the same way as for (57) :

s^\ ft 1

. . (59)

Putting a = 1118 ft. per sec., the value appropriate to 15° C., gives,
for example :

q (ft. per sec.) : 100 200 400

(po — p)/foq*: 1*002 1*008 1-032

A.D.— -2*

42 AERODYNAMICS [CH.

Thus the correction on (58) due to compressibility remains small for
moderately large velocities.

In the above case of motion the dividing streamline is obviously
straight, and collinear with the axis of the tube. Imagine a solid of
revolution, of the shape of an airship envelope, for instance, having
this same axis and situated with its nose at the mouth of the tube.
The pressure in the tube remains unchanged, and indicates a pressure
increase of %pqz occurring at the nose of the body. An airship nose
requires special strengthening to withstand this pressure (cf. Fig. 7).
If the body and tube are tilted with respect to the oncoming stream,
i.e. are given an ' angle of incidence/ the pressure in the tube de-
creases. But it must then be possible
to find a new position for the tube, in
the neighbourhood of the nose of the
body, such that the pressure difference
'in the tube is again Jp#2, for there must
still exist a dividing streamline,
although now it may be curved (Fig.
13). Experiment confirms this con-
clusion.

The point at which the dividing
streamline meets the nose of an im-
fftersed body is called the front

stagnation point. ^The increase of pressure there is known as the
stagnation pressure. ^ The fact that a stagnation point must exist is
of considerable help in constructing curves of pressure variation
round the contour of a body from meagre experimental data.

33. Basis of Velocity Measurement

The undisturbed static pressure of a stream is measured as de-
scribed in Article 27. A combination of a pitot tube and a static
pressure tube, called a pitot-static tube, enables local velocity to be
measured if p is known. For from (58)

FIG. 13. — FRONT STAGNATION
POINT.

? =

(60)

The velocity thus obtained may be corrected, if need be, for com-
pressibility by (59). A concentric form of pitot-static tube is shown
in Fig. 14 ; other designs exist.

Other methods of measuring velocity are readily devised, although
none is so convenient. The present method has a theoretical advan-
tage in determining directly not q but pq2. It is usually the latter

AIR FLOW AND AERODYNAMIC FORCE

quantity that is required to be known with accuracy in Aerody-
namics ; often a comparatively rough knowledge of q itself is
sufficient.

Various problems in connection with the use of pitot-static tubes
are described later ; but a certain limitation may be referred to here.

1=3

FIG. 14. — N.P.L. PITOT-STATIC PRESSURE TUBE.

Putting q = 10 ft. per sec. gives for standard conditions at sea-level
pQ — p =0-119 Ib. per sq. ft. This pressure difference balances a
head of water —0-023. in. only. Gauges (compare, for instance,
Article 7) can be devised to measure such a pressure with high
accuracy, but the required sensitivity makes simple forms unsuited
to rapid laboratory use, owing to various small disturbing factors,
which are usually negligible, beginning to become important. Varia-
tion of temperature, vibration, and slight wear are instances. 1 or
2 per cent, of the above head is a convenient limit to sensitivity. It
follows that the pitot-static tube becomes unsuitable for smaller
velocities, and other means of measurement are then substituted.
Of these, the change in electrical resistance of a fine heated wire due
to forced convection in a stream has proved most convenient.

A pitot-static tube, usually of divided type, is employed on air-
craft to indicate speed. Wherever located within practical limita-
tions, it is subject to disturbance from near parts of the craft to an
extent depending on speed. Especially if fitted to an aeroplane, the
tube can only be tangential to the local stream at one speed. Errors
due to an inclination of 10° amount to 2-3 per cent., depending upon
the type of tube. A mean alignment is adopted, but for the several
reasons stated calibration in place is necessary for accurate readings.

The tube is connected with a pressure gauge of aneroid barometer
type, deflection of a diaphragm of thin corrugated metal moving a

44 AERODYNAMICS [CH.

needle over a scale calibrated in miles per hour. The reading is
termed indicated air speed (A.S.I.), and gives the true speed of the
craft relative to the air at low altitude only. If true speed is required
at considerable altitudes, readings must be increased in the ratio
VT/cr, where cr is the relative air density.

Special forms of pressure tube also exist for aircraft, designed to
permit use of a more robust gauge. Since increase of pressure cannot
exceed Jp#2, except on account of compressibility, the static tube is
replaced by a device giving less than static pressure. This some-
times consists of a single or double venturi tube. Particulars of
Venturis for this and other purposes are given in the paper cited *
(as exposed on aircraft they are not constrained to 'run full').
A formula for the pitot pressure at speeds exceeding the velocity of
sound is given later.

SUBDIVISION OF FLOW PAST BODIES

34. Taking advantage of the outstanding result of Article 30, it
will now be assumed, except where stated otherwise, that the fluid
is sensibly homogeneous
and flows incompressibly.
From Article 3 1 , maximum
velocities must not ap-
proach that of sound in
air. A very useful ex-
pression of the assump-
tion is obtained as follows.

Consider part of the
field of a two-dimensional

FIG. 15.

flow enclosed within any

small rectangle ABCD

(Fig. 15), of sides $x, Xy, u,

v being the components parallel to Ox, Oy of the resultant velocity.

The rate at which fluid mass tends to be exhausted from the rectangle

owing to difference in velocities and densities at BC and DA is

^ } Sjy — p«8y = ~~ 8#8y . Comparing similarly the mass-
es / ox

flow across the sides AB and CD, the rate at which matter is ex-

hausted from the rectangle on this account is -~- S#8y. Now density

* Piercy and Mines, A.R.C.R. & M. 664, 1919.

II] AIR FLOW AND AERODYNAMIC FORCE 45

is assumed to remain constant. Hence :

du dv

This expression is known as the equation of continuity for an incom-
pressible fluid.

35. When a wind divides to flow past an obstacle, such as an air-
ship, held stationary within it, the inertia of the air tends to localise
to the vicinity of the body the large deflections that must occur in
the stream, so that laterally distant parts are little affected. Imagine
a hoop of diameter several times as great as the maximum trans-
verse dimension of the body to be held across the stream, enclosing
the body. The volume of air flowing through the hoop per sec. is
little diminished by the presence of the body, the air flowing faster
to make up for the obstructed area. As the diameter of the hoop is
decreased, this statement becomes less true, but at first only slowly.
In other words, the increase of speed increases as the body is ap-
proached. If there were no friction at the surface of the body, the
speed would reach a maximum there. But (Article 22) the air is
stationary on the surface, and is retarded for some distance into the
fluid. We are concerned with the manner in which such retardation
consorts with the more distant, though still close, increases of speed,
which are often large.

36. Experimental Streamlines

It is always possible to plot the streamlines for a steady motion
from experimental knowledge of the velocity distribution. Fig. 16
has been prepared from actual measurements of the approximately
two-dimensional motion in the median plane of a scale model of an
aeroplane wing of the section shown. The model wing, or aerofoil,
was immersed in a stream whose velocity U and pressure pQ were
initially uniform. Explorations of the magnitude and direction of
the disturbed velocity q were made along several normals to the wing
surface ; values of q sin a/C7 are plotted for the two shown, viz.
SlNl and 52]V2, distance from the surface along either normal being
denoted by n and the angle between q and the normal by a.

The flow across any part of a normal is given by the value of the
integral

\q sin a dn

over that part. Choose a point A± on SlNl through which it is
desired that a streamline shall pass. Evaluate graphically

46 AERODYNAMICS [CH.

ql sin a dn = k, say.
For n small, sin a = 1-0. Now find a point A* on S2NZ such that

I-

O2 O3 O4 0-5 0-6

FIG. 16.

i.e. find the line n = A* (Fig. 16), such that the area OA^A* equals
the area OA ^A^ Similarly, determine points A, A . . . along other
normals. Now there is no flow across the aerofoil contour. There-
fore there is no flow across the curve AAtA2A . . , Hence this
curve is a streamline.

Successive streamlines follow by changing k to k', k* . . . It is

II] AIR FLOW AND AERODYNAMIC FORCE 47

convenient to make k = k' — Tc = k" — fc' = . . ., for then, if the
intervals are sufficiently small, the velocity is inversely proportional
to the distance apart of successive streamlines. A second streamline
is, of course, most conveniently constructed from the first, a third
from the second, and so on.

37. The Stream Function

It would be possible to fit to an experimental streamline a formula
/ (x> y) = constant. The fit would not be so close, however, nor the
original measurements so
accurate as to ensure
obtaining another stream-
line by equating the same
function of x and y to
another constant. In a
motion that is known
analytically, on the other
hand, these difficulties

FIG. 17,

disappear. We then have *
a function of x and yt
fy (#> y)> which, on equat-
ing to any constant, gives corresponding values of x and y for
points lying on one of the streamlines of the motion, fy is called
the stream function of the motion.

Consider a steady two-dimensional motion in the #y-plane. Let
A and B be two points, not on the same streamline ; join them by
any curve (Fig. 17), and let q make an angle a with an element 8s
of the curve. Define the flow across this curve by fy3 — t];A, i.e.

fB .
YB — YA ==\ ? sin a as.

J A

This value is unique, for the flow across AB is independent of the
shape of the curve, being the same as that across any other curve,
such as ACB, joining the points, since otherwise fluid would be com-
pressed within, or exhausted from, the area ACBA.

With A fixed let B move in such a manner that the above flow
remains constant. Then B traces out a streamline, because there is
no flow across its path. If the value of fy (x, y) at A = k, for all
points on the streamline BB',

It follows that the equation to all streamlines is

ty = constant .... (62)

48 AERODYNAMICS {CH.

the constant changing from one to another. A definite value is
assigned to the constant of a particular streamline by agreeing

to denote some chosen
streamline by fy (x, y) = 0.
A question of sign is
involved ; the increment
of <|j is taken as positive if
the flow is in a clockwise
direction about the origin,
but sign is determined
generally by (63) below.

38. Let A and D be
adjacent points on two
streamlines : = k and

FIG. 18.

<Jj = fc + Jty. The co-or-
dinates of A are x and y, those of D x + %%> y + Sy. From Fig.
18, the flow across AD = that across ED less that across AE.
Hence if u, v are the components parallel to Ox, Oy, respectively,
of the velocity q,

Now 8^ is the total variation of a function of the two independent
variables x and y. It is assumed that the partial derivatives dfy/dx
and 3^/3^ are also continuous functions of x and y. It is shown in
text-books on Calculus that then

Hence :

-s
ty

(63)

As an example, suppose <]* = Uy, where U is a constant. From
(63) u = [7, v = 0, and the flow evidently consists of uniform
motion at constant velocity U in the direction Ox. Putting
ty/U = 0, 1, 2 . . . gives a series of streamlines all parallel to Ox
and spaced equally apart. Again, consider the flow fy = Qy8, where
C is a constant. Putting <J*/C = 0, 1, 2 . . . again gives streamlines
parallel to 0#, but at a decreasing distance apart (Fig. 19). From
(63) u = 2Cy, v = 0, and we recognise the flow as including that of

AIR FLOW AND AERODYNAMIC FORCE

y/c*6

y/os

Wo*

y/c=3

Article 24, where 2C = U/h,
and there are certain restric-
tions on the area occupied by
the flow.

39. Circulation and Vorticity

So far we have dealt with
the line integral of the normal
velocity component across a
curve drawn in the field of
flow. The line integral of the
tangential velocity component
once round any closed curve is
called the circulation round

that circuit and is denoted by K. If Ss is an element of
length of the closed curve of the circuit, q the velocity, and a
the angle which q makes with Ss,

FIG. 19.— STREAMLINES FOR UNIFORM
SHEARING.

K = Jc q cos a ds .

(64)

There is again a question of sign, and this is taken as positive if K
has a counter-clockwise sense.

Let us calculate the circulation $K round the small rectangle
ABCD (Fig. 20), of sides 8x, Sy. The sides AD and CB
together contribute to counter-clockwise circulation an amount

(du \ Su

u + Y 8y\ 8% = - — S#Sy. Similarly DC and BA together

^i
contribute — Sx Sy.

Hence :

8x8y dx

!**
9y

^vtl^Sx

The finite limit to which
the left-hand side tends as
the area decreases is called
the vorticity at the point
and has the symbol £.
Thus :

£=4!-!:? - (66)

FIG. 20.

In words, the vorticity of an element is the ratio to its area of the

50 AERODYNAMICS [CH.

circulation round its contour. Choose the element as circular, of
radius r, and so small that its angular velocity G> can be considered
constant. Then K is due to G> alone. Writing S for area,

SK = 2-nr . <or ; dK/dS = 2to
or

Thus the vorticity of an element is twice its angular velocity.

In the first example of Article 38, where u = £7, a constant, and
v = 0, we now have from (65) £ == 0 everywhere ; the elements of
fluid are devoid of spin. For the second example, u = 2Cy, v = 0
and (65) gives £ = — 2C, a constant, or there is a uniform distribu-
tion of vorticity. Applying the latter result to the motion of Article
24, £ = — C7/A. Imagine the moving plate in Article 24 to be
started from rest. Initially u = v = 0 and the fluid is devoid of
vorticity, but after a sufficient time a uniform distribution of vorti-
city is generated, arising from the boundary condition of zero slip
and the action of viscosity. We are thus able to trace the generation
of the vorticity to viscosity. If the pressure had acted equally in all
directions, it would have exerted no couple on any element of fluid
which, being originally devoid of vorticity, would have remained so.

40. Extension of Bernoulli's Equation

We are now in a position to prove a theorem of practical impor-

tance in connection with flow that is sufficiently distant from bodies

and other boundaries. A distribu-
tion of vorticity is assumed to
exist, but tangential components of
stress are neglected.

Consider the fluid element ABCD
(Fig. 21), bounded by two adjacent
streamlines and the normals thereto.
Let the radius of curvature, assumed
large, of the streamline AB be R.
Let s denote length measured along
AB, DC, and n denote length
measured along either of the nor-
mals towards the centre of curva-
ture. Let q be the velocity along

AB. Tangential components being neglected, the pressures act

normally to the faces of the element,
The element exerts a centrifugal force p$s8nq*/R which, the flow

RADR

II] AIR FLOW AND AERODYNAMIC FORCE 61

being steady, is balanced by a force due to the difference of the
pressures on the faces AB, CD, i.e. by the force — 8s8n(Sp/Sn)
(Article 28). Hence :

Now calculate the circulation 8K round the element. There is no
flow along the normals ; hence :

. . (i)

From the figure —

CZ> __ CD _ __ 8n

AB ~~&T ~~l ~~~R'
Substituting in (i)

-sT .-

dn R dn

The last term is evidently negligible compared with the others.
Hence, finally :

- q dq

Multiply both sides of (67) by — pj and substitute from (66) for
IjR. (67) becomes

Now p + Jpy* is the pitot head (Article 32). Thus, across the
streamlines the pitot head has a gradient proportional to the product
of the velocity and the vorticity, provided tractions can be neglected.
If, on traversing a pitot tube across a field of flow, the pitot head
remains constant, then the flow is devoid of vorticity so far as it
is explored.

41. Irrotational Flow

An irrotational motion is one in which it is everywhere true that
% = 0. Where velocity gradients exist, this condition usually

AERODYNAMICS

[CH.

appears as an ideal which is not exactly attained by a real fluid, but
many motions of great Aerodynamic interest approximate closely to
the irrotational state. These are discussed theoretically in later
chapters. Meanwhile, we note that the theorem of the preceding
article leads, as described, to a convenient method of investigating
experimentally whether a given flow, or what part of it, is approxi-
mately irrotational.

42. Subdivision of Flow Past Bodies

It will now be shown, on experimental grounds, as will be proved
theoretically in a later chapter, that Aerodynamic types of flow can

0-0

O Ol O2 O-3

l-O

be separated into two parts : an outer irrotational motion and an
inner flow characterised by the presence of vorticity. For this pur-
pose a particular, but typical, case will be described in some detail.
The flow selected is that above the aerofoil of Article 36. This aero-
foil was small, having a chord c (length of section) of l£ in. It was
set at an incidence (angle made with the oncoming stream by the
common tangent to its lower surface) of 9-5° in an initially uniform
air-stream of velocity U = 41 ft. per sec. and pressure p0. The
undisturbed stream was verified to be sensibly irrotational by track-
ing across it a pitof 'tube, the pitot head being found to be constant.

II] AIR FLOW AND AERODYNAMIC FORCE 53

The same two normals, S1N1 and S2N2) distant c/3 and 2c/3, respec-
tively, from the leading edge, were selected for study as those for
which the variation of velocity (q) has been given in Article 36. What
will now be described is the variation that was found along them of
pitot head and static pressure (p). In Fig. 22 the first of these is
given in the form :

; to

ht then, being the loss of pitot head caused by the model ; the second
is conveniently expressed as (p — p0)/pU*, both quantities being
non-dimensional .

It is seen from the figure that the aerofoil causes negligible change
of pitot head beyond n = 0-04c for the upstream and n = 0-2c for
the downstream normal. Beyond these limits the stream is con-
cluded to be irrotational, approximately ; just within them the
velocity gradients are not large and the tractions may be expected
to be small, so that, from (68), we infer vorticity to be present. By
traversing a fine pitot tube along a number of other normals, or lines
across the stream, a number of similarly critical points for pitot
head can be found. A line drawn through all such points forms a loop
which wraps itself very closely round the nose of the model (where a
special form of pitot tube is
necessary for detection),
widens as the trailing edge
is approached, and finally
marks out a wake behind the
aerofoil. Fig. 23 shows the
wake located in this way behind another aerofoil set at smaller
incidence. The complete loop may be called, for short, the pit<
boundary * and is one way of marking out an internal limitation
to irrotational flow.

The pressure decrease pQ — p builds up along the normals as the
aerofoil is approached to maxima at the pitot boundary. Actually
there was no reason to measure the pressure and velocity separately
in the outer irrotational region except as a check, for here the one
can be calculated from the other by Bernoulli's theorem. The
maximum pressure changes generated at the pitot boundary are
transmitted without further variation along normals to the aerofoil
surface. This important point is clearly seen from the pressure

* For further illustrations see Piercy, Jour, Roy. Afro, Soc., October 1923.

54 AERODYNAMICS [CH.

curve for the more downstream normal, the three readings nearest
the surface being

n\c = 0-047 0-127 0-173

£^~ = 0-310 0-308 0-312

A pressure change of 0-47 pC72 is transmitted to the surface along the
more upstream normal. Within the pitot boundary adjacent to the
aerofoil the velocity (Fig. 16) and the pitot head (Fig. 22) fall away
rapidly. The first, as already seen, vanishes on the surface ; the
second decreases from p0 + ipt^2 to p', the value of the static pres-
sure on the pitot boundary opposite the position round the contour of
the aerofoil considered.

The chain-line curve (Fig. 22) gives a wider view of the manner in
which the static pressure drop is built up.

An aeroplane was fitted with wings of the shape of the aerofoil,
and some measurements were made in flight. These showed the
velocities and pressures, non-dimensionally expressed, to be different
from those observed with the model but not greatly so. The pitot
boundary was found to be much closer to the full-scale wing surface
than it was to the model surface when expressed as a fraction of the
chord.

43. The Boundary Layer

From the many experiments which have been made on lines simi-
lar to the foregoing, we draw the following preliminary conclusions
regarding motions of Aerodynamic interest past bodies :

1. There exists an outer irrotational flow.

2. This is separated from the body by a sheath of fluid infected
with vorticity arising from the boundary condition of no slip and
the action of viscosity. This sheath or film of fluid increases in
thickness from the nose to the tail of the body, but is nowhere thick
and is called the Boundary Layer. It merges into the wake.

3. Changes in static pressure are built up in the outer flow, related
to the velocity changes there by Bernoulli's equation, and are trans-
mitted to the surface of the body through the boundary layer.

AERODYNAMIC FORCE AND SCALE

44. The Aerodynamic force on a body is that resultant force on it
which is due solely to motion relative to the fluid in which it is

II] AIR FLOW AND AERODYNAMIC FORCE 55

immersed. Thus forces acting on the body due to gravity, buoyancy,
etc., are excluded. Aerodynamic force arises on the body in two
ways : (a) from the static pressures over the surface, sometimes
called the normal pressures ; (b) from a distribution of skin friction
over the surface.

Consider, for example, an aeroplane wing of uniform section. Let
8s denote the area, per unit of span, of an element of the contour of

IVeesui
increase

Pressure
decrease

FIG. 24.-

-EXPERIMENTAL PRESSURE DISTRIBUTION ROUND SECTION OF AEROFOIL,
SHOWING INTEGRATION OF PRESSURE DRAG AND LIFT.

the section at S (Fig. 24), and 0 the angle which the normal SN at
8s makes with SL, the perpendicular to the direction of the relative
undisturbed wind. For convenience, subtract from the pressure
acting on the surface the static pressure of the oncoming stream, and
let p be the normal component of the remainder and F the tangential
component. The variation of p is shown by the dotted line in the
figure. The force on the element is compounded of p.8s, outwardly
directed along SN and F.Ss, perpendicular thereto. For simplicity
we shall assume the flow to be two-dimensional, so that F has no
component parallel to the span, p and F vary from point to point
over the wing, leading to a variation of force from one element to
another in both magnitude and direction. To obtain the resultant
force we require to effect a summation of the forces on all elements.

56 AERODYNAMICS [CH.

Evaluation is usefully simplified in the following way : compon-
ents of the resultant force are determined parallel to SL drawn
perpendicular to the relative wind and to the aerofoil span, and SD
in the direction of the relative wind. The first component is
the lift, the second the drag. It will especially be noted that the
Aerodynamic lift of a wing, unlike the static lift of a gas-bag, is not
constrained to be vertical, nor even does its direction necessarily lie
in a vertical plane ; it is perpendicular to the span of the wing and
also to the relative wind, and is taken as positive if it is directed
upward when the wing is the right way up.

Denoting lift by L and drag by Z), we have for 8s

8L = (p cos 0 + F sin 0) 8s
SD = (- p sin 6 + F cos 6) Ss

Now Ss sin 0 and Ss cos 0 are the projections of Ss perpendicular and
parallel to SL. Hence, if the aerofoil section is drawn accurately to
scale and all points on the contour at which p is known are projected
upon a line perpendicular to SL (i.e. upon a line parallel to the un-
disturbed relative wind) and p is set up normally to this line, the
area enclosed by the curve obtained by joining the points, completed
so as to represent the whole contour, is proportional to that part of
the lift which is due to p. If similar projections are made along a
line parallel to SL, and p is set up normally to this line and a closed
curve is obtained by joining all points and completing so as to include
all positions round the contour, the net area enclosed by the curve is
pioportional to the contribution to drag by p. As regards drag, the
simplest curve found for an aerofoil is of figure-of-eight form, one
loop of which is positive and the other negative ; the net area is con-
veniently obtained by tracing the point of a planimeter round the
diagram in a direction corresponding to one complete circuit of the
aerofoil contour. The contributions of skin friction to lift and drag
are similarly determined, but the directions of projection are inter-
changed. The sense of F depends upon that of the velocity
gradient with which it is associated. However, the correct sense is
easily decided by inspection.

45. An example of the variation of p round the median section of
an aerofoil at a certain angle of incidence, experimentally determined
at a certain speed, is given in Fig. 24. Curves are also shown
obtained by projection perpendicular to and in the direction of the
oncoming stream, the areas under which are proportional to the lift
and drag per foot run of the span at the median section, the area
ABC giving negative contributions to drag. Apart from scientific

II] AIR FLOW AND AERODYNAMIC FORCE 57

interest, investigations of distribution of force are of technical im-
portance, especially in the case of aerofoils, providing data essential
to the design of sufficiently strong structural members of minimum
weight for the corresponding aeroplane wing. Such analysis is
usually required at several angles of incidence. Since the pressure
will most conveniently be found at the same points round the contour
for all incidences, labour is saved by projecting along and normal to
the chord of the wing, resolving subsequently in the wind direction
and perpendicular thereto. Graphical processes of integration con-
venient for bodies other than wings will be left for the reader to devise.

46. Some limiting cases may be mentioned. When a fluid flows
through a straight pipe or past a thin flat plate at zero incidence —
i.e. parallel to the oncoming stream — the drag must be wholly fric-
tional. Such drag is small with air as fluid. At the other extreme,
the drag of a thin flat plate set normal to the undisturbed stream
must arise wholly from unequal distribution of pressure. This drag
is comparatively large, but is less than that of a cup-shaped body
with the concavity facing the direction of flow, as instanced by a
parachute. Referring to aerofoils, the contribution of skin friction
to lift is negligible. The area enclosed by the negative drag loop of
the projected pressure curve (e.g. ABC of Fig. 24) may approach
that of the positive loop, when the contribution of the pressures to
drag will be small. For 'this condition to be realised, the flow must
envelop the back of the body closely, i.e. without ' breaking away '
from the profile. Negative drag loops are absent from the normal
plate and very small for the circular cylinder.

A quantity of significance descriptive of an aerofoil is the ratio of
the lift L to the drag D, i.e. L/D. Since D = L ~- L/D, for a given
lift the drag is smaller the greater L/D. Considering a flat plate at
any incidence a and neglecting skin friction, and writing P for the
total force due to variations of pressure over the two surfaces, we
have L = P cos a, D = P sin a. However P varies with a, L/D =
cot a. Values given by this formula must always be excessive,
greatly so at small incidences when the neglected skin friction
becomes relatively important. Nevertheless, at flying incidences
the L/D of a wing, skin friction included, greatly exceeds cot a. A
wing having also essentially more lifting power than a flat plate, this
comparison is often 'given as illustrating the superiority of the
aerofoil over the flat plate for aeroplane wings. The advantage is
seen to arise from the pressure distribution round the forward part
of the upper surface of the aerofoil, providing positive lift and nega-
tive drag.

58 AERODYNAMICS [CH.

As a matter of experiment it is found that the pressure drag of a
carefully shaped airship envelope almost vanishes, although the
pressure varies considerably from nose to tail, and the drag is almost
wholly frictional ; it may amount to less than 2 per cent, of the drag
of a normal disc of diameter equal to the maximum diameter of the
envelope. The example illustrates the great economy in drag which
can be achieved by careful shaping, a process known as fairing or
streamlining. So exacting is this process that it pays to shape the
contour of a wing, strut section, engine egg, or other exposed part
of an aircraft by some suitable formula, instead of using french
curves, so as to avoid sharp changes of curvature which, although
scarcely apparent to the eye, may increase drag considerably.

47. Rayleigh's Formula

Further investigation of how Aerodynamic force depends upon
shape is left to subsequent chapters. The knowledge required for
practical use will result partly from theory and partly from experi-
ment. For both lines of enquiry we need to establish a proper scale
in terms of which Aerodynamic force may be measured. For this
purpose we keep the geometrical shape of the body and its attitude
to the wind constant, but allow its size to vary ; in other words, we
consider a series of bodies of different sizes made from a single
drawing, immersed, one after another, at the same incidence in a
uniform stream of air. Geometrical similarity must include rough-
ness of surface, unless effects of variation are known in a given case
to be negligible ; a caution is also necessary against tolerating any
lack of uniformity in the oncoming stream. But the velocity of the
stream may vary and also the physical condition of the air ; in fact,
the bodies may be supposed immersed in uniform streams of different
fluids, liquid or gaseous. But it is assumed for simplicity, and as
representing a common condition in Aerodynamics, that maximum
velocities attained are small compared with the velocity of sound in
the fluid concerned, so that compressibility may be neglected.

Preceding articles have shown that the Aerodynamic force A
arises from pressure variation and skin friction. The pressure will
depend upon the density p and the undisturbed velocity U. The
skin friction has been seen to depend upon U and the viscosity jju
Comparing different fluids, or air in different states, the general effect
of viscosity depends on the ratio of the internal tractions to the
inertia, which is proportional to p. Hence it is convenient to sub-
stitute for IJL the quantity

v-jji/p . . . . (69)

II] AIR FLOW AND AERODYNAMIC FORCE 59

called the kinematic coefficient of viscosity, whose dimensions are
(cf. Article 23) M/LT -r M/L* = L*/T. The Aerodynamic force,
since it results from the surface integration of pressure and skin
friction, will also depend upon the size of the body, which is specified
by any agreed representative length /, because the geometrical shape
is constant.

It is concluded, then, that :

A depends upon p, U, I, v . . . (70)

and on nothing else. This conclusion, which is essential to the
investigation, can be arrived at in other ways ; e.g. by appeal to
simple experiments. Thus, if a bluff shape, such as a normal plate,
is moved by hand through air, drag can be felt to depend upon size
and velocity. If it is then moved through water, a great increase
occurs mainly as a result of increased density. Moving the plate
finally through thick oil instead of water shows that drag also
depends upon viscosity, for density need scarcely have changed.
The importance of the more careful consideration that we have given
to the question lies in the assurance that no important factor has
been omitted.

It is desired to obtain a general formula for A , connecting it with
p, U, I, v. This may contain a number of terms, any one of which
can be written in the form :

?pUqlrv5 (71)

Now A, being a force, has the dimensions of mass x acceleration,
i.e. ML/T2. The principle of homogeneity of dimensions asserts *-!;;:[
all terms in the formula for A must have the same dimensions.
Writing (71) in dimensional form :

ML (M\*(L\< /L-V

r» ViV \r/ \T/

For the dimensions of the term to be ML/T*, it is required that
on account of the M's, p = 1,
on account of the L's, — 3p + q + r + 2s = l,
on account of the T's, — q — s = — 2,
giving

/> = !

q =y = 2 — s.

Hence the formula for A is :

A - 2p[/2~'/2~*vs

60 AERODYNAMICS [CH.

or A =P W. /, . • • (72)

where f(Ul/v) means some particular function of the one variable
OT/v.

This important relationship is the simplest case of Rayleigh's
formula. The investigation equally leads to

\ V

.... (72a)

an alternative form of particular use where changte of fluid is involved.
It will be noted that Aerodynamic force cannot vary with the area
of the body or the square of the velocity exactly unless it is indepen-
dent of viscosity, which is absurd.

48. Reynolds Number — Simple Similar Motions

The quantity £7//v is called the Reynolds number after Osborne
Reynolds, who first discovered its significance, and is written R.
Writing (72) as

.... PD

we have, on the left-hand side, a coefficient of Aerodynamic force
.vnose value for any shape of body and value of R can be found if
required by actual measurement.

Still keeping shape constant, let us investigate what similarity
exists in the flow of different fluids at different velocities past bodies
of different sizes, subject to the restriction that R remains constant.

Considering any particular position in the field of flow past the
particular shape, the method of Article 47 readily gives for any
velocity component there, for instance u,

Hence, from consideration of velocity components at right angles at
geometrically similarly situated points, called corresponding points,
one in each of a series of fields of flow past bodies of the same shape
(and attitude) at the same Reynolds number, the resultant velocity
there is the same in direction. Since this is true of all sets of corres-
ponding points, the streamlines present the same picture, though to
different geometric scales. The magnitude of the velocity at corres-

Il] AIR FLOW AND AERODYNAMIC FORCE 61

ponding points oc U ; and the pressure oc pt/f, as may be shown
directly.

It follows that at corresponding points on the contours of the
bodies the pressure oc pt/2 and the skin friction oc y.U/1 oc pvt///, and
that part of A resulting from pressure variation oc pt/2/2, while that
part due to skin friction oc pvl// oc p£/2/2, since v oc Ul, because R is
constant. Hence A oc pC72/2, or the left-hand side of (73) is constant.

Example : if also the fluid is constant, show that A is constant.

The foregoing assumes the motions to be steady. Now let them
have frequencies ~ (dimensions : 1/T). With frequency assumed
to depend only on p, U, I, v, the method of Article 47 gives :

~ = -

While R remains constant, ~ oc U/l in periodic motions. If also
the fluid be given so that v, and therefore Ul, remain constant,
~ oc U* oc l//a. The streamlines pass through the same sequence
of transient configurations but at different rates ; if cinema films
were taken of the motions, any picture in one film would be found
in the others, but it would recur at a different, though related, fre-
quency. Similarity of streamlines, etc., as described above, then
occurs at the same phase. The result : A oc p[72/2 is now true of
the Aerodynamic force at any phase and also of the mean value, with
which we are usually concerned.

The motions considered in this article provide an example of what
are termed dynamically similar motions. Constancy of the left-hand
side of (73) is also found by experiment for R constant when the
bodies produce flow that varies rapidly in an irregular manner.

49. Aerodynamic Scale

When the Reynolds number changes, there is no reason to expect
the coefficient of Aerodynamic force to remain constant, and it is
found to vary, sometimes very little through a limited range of R,
sometimes sharply, depending upon the shape of the body (or its
attitude) and the mean value of R, Now, if by a series of experi-
ments or calculations we obtain a number of values of A for a given
shape, work out the coefficients and plot these against R, it is clear
from Article 48 that all coefficients will lie on a single curve. This
curve is the graphical representation of f(R) through the range
explored.

Fig. 25 gives as an example the variation of (drag -f- p£72/2) with
R for long circular cylinders set across the stream. In order to fix

62 AERODYNAMICS [CH.

the numerical scales, it has been chosen quite arbitrarily to use the
diameter of the cylinder in specif yingR and the square of the diameter
for /», but the drag then relates to a length of the cylinder equal to its
diameter. The full line results from a great number of observations.
These are not shown, but they fit the curve closely, though a cluster
of points round a particular Reynolds number may include great

l'\~f

O-8

^fp

O-Zt

/
/

/
/
/
/

^£O

O15

Ol
O05

x "

"^/
i\

~~^^

X-

-X

m '

\~t
u

/ \

1O \OZ 103

1O5 1O6

FIG. 25. — DRAG OF LONG CIRCULAR CYLINDERS SET ACROSS STREAM AND FREQUENCY
OF FLOW IN WAKE (/ «== DIAMETER).

variation in, for instance, diameter. The rapid rise of drag at R =
10e flattens again at 1-3 x 10« with a value of about 0-3 for the co-
efficient. The broken line gives the variation of frequency, the flow
eddying for R > 100.

Similar success has been obtained experimentally in many other
cases, and we conclude that the theory of Article 47 can be accepted
with confidence. When observations at constant Reynolds number
disagree with one another, the cause is to be sought in the particular
circumstances of the experiments ; if geometrical similarity is truly
realised and velocities are demonstrably too small for appreciable

II] AIR FLOW AND AERODYNAMIC FORCE 63

compressions and expansions, the cause may be traced to consider-
able variation of unsteadiness in the oncoming streams.

Finally, it becomes evident that, with moderate velocities, the
Reynolds number provides a proper scale for Aerodynamic motions.
Circumstances in which this scale is not suitable are described in
the following articles.

The principle of dimensional homogeneity is often employed to
express in a rational formula the results of a series of experiments
on a given shape. The process usually depends upon discovery of a
constant index for one of the variables, although this restriction is
not necessary. It should carefully be noted that such formulae apply
only through the range for which they have been shown to hold ;
large errors often result from extrapolation. Thus, such formulae
amount to no more than a convenient mental note of the results from
which they are derived ; they constitute merely an approximation
to part of the f(R) curve for the shape concerned.

The outstanding practical significance of general formulae such
as (72) is to establish the basis on which single experiments on scale
models of aircraft or their component parts should, if possible, be
carried out. Provided the model is tested at the same Aerodynamic
scale, experimental measurements are accurately related to corre-
sponding quantities at the full scale ; otherwise corrections ire
necessary. The proviso can by no means always be satisfied even
when the gauge of Aerodynamic scale is simply the Reynolds
number. The more complicated formulae completing this chapter
will show that the Reynolds number alone is often insufficient ; the
position then becomes more difficult and experiment requires
planning with judicious care.

49 A. Rayleigh's Formula — High Speeds

If the compressibility of the air cannot be neglected, its modulus
of bulk elasticity E must be admitted and the typical term in the
new formula for Aerodynamic force becomes

ftVWE* ... (i)

The dimensions of E are M/LT*, and the method of Article 47 gives
(M) I = p + t

(L) l = —3p + q+r + 2s — t
(T) -2 = - q - s - 2t,
i.e.,

p = 1-*

q = 2 — s — 2t

r = 2 — 5,

64 AERODYNAMICS [Cfi.

giving the result —

By Article 20, E — pa2, where a is the velocity of sound, i.e.
pt/2/£ = (U/a)*. The ratio t//0 is called the Mach number and de-
noted by M, and the formula becomes finally

A = 9UW.f(R, M) ...... (73A)

There being five unknowns and only three dimensional equations
to relate them, the new function has two arguments ; A depends
upon both and, theoretically, this dependence cannot be separated.
From calculation of the stagnation pressure, an inference has been
made in a preliminary way that compressibility can be ignored for
speeds little greater than 250 m.p.h. at low altitude, i.e. for values
of M less than J, and considerably greater values produce in some
cases only negligible effects on A . Formerly, the airscrew provided
almost the only occasion calling for a formula of the type (73A),
speeds towards the tips of their blades being so high as to make M
approach unity. In modern Aeronautics, however, the importance
of the formula is much wider. Considering, for example, a strato-
spheric aeroplane flying at the moderate indicated air speed of
200 m.p.h. at an altitude of 40,000 ft., where the relative density
of the air is J, U = (22/15) . (200/Vi) = 58? ft- P^r sec., whilst a is
reduced by the low temperature to the value 975 ft. per sec., giving
M > 0*6 for every part of the aeroplane. Now in full-scale flight
at little more than this Mach number the effect of varying M may
be much more important than that of varying R. Thus, while
(72) can still be relied upon in a great variety of practical circum-
stances, the occasions on which it is superseded by (73A) are
multiplying.

For two motions 1 and 2 to be dynamically similar, both R and
M must be the same, leading to

For a dynamically similar experiment on a model of an aircraft
it will be plain from the next chapter that the power required to
produce the artificial wind is economised chiefly by reducing the
speed. But this would involve employing very cold air. In

II] AIRFLOW AND AERODYNAMIC FORCE 66

these circumstances and in view of the labour involved, the task of
constructing a data sheet such as Fig. 25, which would now embrace
a series of curves for a body of given shape, is abandoned, and
experiments on the effect of high Mach numbers are usually carried
out with no more than the precaution of avoiding very small
Reynolds numbers.

498. Some Other Conditions for Similitude occurring in Aerodynamics

(1) When a seaplane float or flying-boat hull moves partly im-
mersed in water, waves formed cause variation of pressure over
horizontal planes due to the weight of the heaped liquid. Thus
gravity comes into the problem of similarity. Approximate treat-
ment ignores air drag of parts projecting above the surface and also
surface tension. Then p, C7, /, v refer to the water only and, with g
added, we write any term in the formula for drag as

Dimensional theory at once gives :

(M) \=p

(L) i==-3p + q + r

(T) — 2 = — q — s — 2t
whence p = 1

q = 2 — s — 2*
r = 2 — s + t

and the term becomes

leading to the following formula for drag :

. (73B)

The drag is made up of two parts : (a) a part akin to Aerodynamic
force but modified by (b) wave-making resistance, which again is
modified by (a). U*/gl is called the Froude number, F.

For dynamical similarity both arguments of the function must be
kept constant. For change of size the second argument gives Ucc<\/l
since g is practically constant, and then by the first v oc ^/l*, i.e. the
fluid must be changed when Doc pvfoc p/8. A change from water is not

A.D. — 3

66 AERODYNAMICS [CH.

convenient, however, and it has been found sufficient, as originally
suggested by Froude, to assume the two kinds of resistance to be
independent of one another, i.e. to write (73B) as

D = Pl7«/« /,(*) + /, 1 . . . * (73C)

This is convenient in regard to the wave-making resistance, because
a model of scale e can be towed in a ship tank at the low correspond-
ing speed : U^/e, where U is the full-scale speed.

One ' ship ' tank (U.S.A.) is 1980 ft. long, 24 ft. wide, and 12 ft.
deep, with a maximum towing speed of 60 m.p.h. Another tank
(R.A.E.) has rather more than one-third these dimensions, with a
maximum speed of 27 m.p.h. In the latter a ^th scale model of a
large hull is feasible, when its maximum model speed would corres-
pond to 81 m.p.h. full scale.

The wave-making resistance is assessed by subtracting from the
total drag measured an estimated Reynolds resistance. The wave-
making resistance is simply related to that under full-scale condi-
tions, to which the Reynolds resistance is added after correction for
rhange of scale.

(2) Froude's law of corresponding speeds reappears, unconnected
with wave-making, in wind-tunnel tests on unsteady motions of air-
craft. The subject is discussed under Stability and Control, but
a simple example will shortly be provided by the ' spinning tunnel/

49C. The airscrew is a twisted aerofoil, each section of the
blades moving along a helical path defined by the radius, the
revolutions per second n, and the forward speed U. To secure
geometrical similarity in experiments on airscrews of different sizes,
each made from the same drawing, it is therefore necessary that
U/nl be constant. Thus a third argument must be added to (73A).
The diameter D is chosen for convenience to specify /, and the non-
dimensional parameter U\nD is given the symbol /. It is also con-
venient to replace U as far as possible by n. Now n1!)4 has the same
dimensions as E71/1, and the formula becomes —

A==9n*D'.f(RtM,J). . . (73D)

Derivation from first principles on the assumption that A depends
on p, £7, /, v, E and n presents no difficulty. But it will now have
become apparent that formulae even more complicated than (73D)
can be constructed from dimensional considerations almost by
inspection.

II] AIRFLOW AND AERODYNAMIC FORCE 67

The following extension of Table III relates to the standard
atmosphere and gives approximate values of various quantities
which are constantly required in calculations of Aerodynamic scale.

TABLE III A

Altitude
(ft.)
-» 1000

I/*
(ft'/sec.)"1
+ 1000

I/a

(ft/sec.)-1
X 1000

1-00

6-4

0-89

0-86

1-29

6-0

0-93

0-73

1-37

1-68

3-8

0-96

0-61

1-63

2-24

2-9

1-00

0-495

2-02

3-33

2-0

1-03

0-39

2-57

6-37

1-2

1-03

Chapter III
WIND-TUNNEL EXPERIMENT

50. Nature of Wind-tunnel Work

The calculation of Aerodynamic force presents difficulties even in
simple cases. Great progress has been made with this problem, as
will be described in subsequent chapters, and designers of aircraft
now rely on direct calculation in several connections. Theoretical
formulae are improved, however, by experimentally determined cor-
rections that take neglected factors into account, while other formulae
are based as much on experiment as on theory. Yet many effects of
change of shape or Reynolds number are of so complicated a nature
as entirely to elude theoretical treatment and to require direct
rfieasurement. Measurements can be made during full-scale flight
by weighing, pressure plotting, comparison of performance, etc. This
method is employed occasionally, but is economically reserved where
possible to the final stages of investigations carried out primarily
on models made strictly to scale. Thus model experiment, which
formerly provided the whole basis of Aerodynamics, apart from the
theoretical work of Lanchester in England and Prandtl in Germany,
still occupies an important place.

In early days of the science, models were sometimes studied out-of-
doors when flying freely (cf. Lanchester 's experiments), suspended
from a balance in a natural wind (Lilienthal), during fall from a
considerable height (Eiffel), or towed. Calm days are few, however,
and unsteadiness of winds was soon found to cause large errors, so
that experiments came to be carried out in laboratories. In the
Whirling Arm method (Langley and others), models attached to a
balance were swung uniformly round a great horizontal circle ; a
disadvantage, additional to mechanical difficulties arising from cen-
trifugal force, lay in the swirl imparted to the air by the revolving
apparatus and the flight of models in their own wakes after the
first revolution. Experiments are now nearly always made in an
artificial wind generated by or within a wind tunnel. This method
was introduced during the second half of the nineteenth century and
wind tunnels were built in various countries during the first decade

CH. Ill] WIND-TUNNEL EXPERIMENT 69

of the present century. A matter of great historic interest is that
the Wright Brothers carried out numerous experiments in a diminu*
tive wind tunnel, less than 2 square feet in sectional area, in prepara*
tion for their brilliant success in the first mechanically propelled
aeroplane, which flew in 1903. The tunnel method of experiment
has since been developed to a magnificent degree.

The artificial wind should be steady and uniform, for otherwise
superposing a velocity cannot change the circumstances of experi-
ment exactly to those of flight through still air. Tunnels can be
designed to achieve a fair approximation to this requirement.
Through the part of the stream actually used for experiment, the
maximum variation of time-average velocity need not exceed
± 1 per cent, and the variation of instantaneous velocity at any one
point, though more difficult to suppress, can be reduced to ± 2 per
cent. This standard of steadiness may be relaxed for experiments
in which it is not of prime importance. The wide range of modern
experiment has led on economic grounds to the evolution of several
specialised forms for the wind tunnel, as will be described, although
in a small Aeronautical laboratory a single tunnel must serve a
variety of widely different uses.

In elementary Aerodynamics it is advisable to carry out many
experiments* which mathematical treatment renders unnecessary
in a more advanced course, but there still remains unlimited scope
for wind-tunnel work on scientific matters in which analysis is of
little avail or particularly complicated. Questions of this nature
will appear as the subject proceeds, and it will only be remarked
here that their investigation invites originality of method and
ingenuity in the design of special apparatus.

Another and equally important domain of model experiment
lies in direct application to specific designs of aircraft. The Aero-
dynamic balances and other measuring apparatus surrounding the
working section of a tunnel have usually been installed with this
purpose primarily in view.

A more or less complete model of an aircraft can be suspended
in a wind-tunnel stream of known speed and its reaction measured.
It can be pitched, yawed, rolled about its longitudinal axis, or
oscillated in imitation of a variety of circumstances arising in free
flight, and its response accurately determined. A special technique
described in a later chapter enables due allowance to be made for
the limited lateral extent of the stream. Yet with every precaution

* A programme of experimental studies requiring only simple apparatus is given in
a companion volume to this book.

70 AERODYNAMICS [CH.

the interpretation of the observations in terms of full-scale flight
is attended with uncertainty. Two outstanding reasons are as
follows. Experiments on complete models, even in national
tunnels, can only cross the threshold of large full-scale Reynolds
numbers, and fall far short in more modest tunnels. Secondly, the
initial turbulence remaining in an artificial wind is sufficient to
produce a marked difference in some connections from flight at the
same Reynolds number.

The first difficulty can be circumvented in the case of small
component parts of an aircraft by employing enlarged models ; for
test in an artificial wind of normal density they would be larger than
full-scale. The drag of the complete aircraft is then built up from
piecemeal tests on its parts. A new problem introduced is to
determine how each part will affect a neighbouring part or one to
which it is joined. Such mutual effect is called interference and
becomes familiar in wind-tunnel work, for in principle it enters into
all experiments in which a model is supported in the stream by
exposed attachments. The same device may be applied to wings
and tail-planes by testing short spanwise-lengths of large chord
under two-dimensional conditions. The consequent problem in this
case is the change from two- to three-dimensional conditions and
is left to calculation. For reliable data on wings at greater inci-
dences or on long bodies, there is no alternative to large or costly
wind tunnels except flying tests.

The above expedients leave the second main difficulty still to be
faced, viz. the effect of initial turbulence. This question is many-
sided and its consideration must be deferred, but there is evidently
need to ascertain by suitable tests the degree of turbulence character-
ising the particular tunnel employed.

Finally, fast aircraft are considerably affected, especially at
high altitudes, by the compressibility of the air. It was found in
the preceding chapter that for dynamical similarity under these
conditions both the Reynolds and Mach numbers require to be
maintained. Tunnels capable of realising even moderate Reynolds
numbers at high speeds are particularly expensive to construct and
operate, and experiments are usually carried out in small streams,
Reynolds numbers being ignored, and the effects of compressibility
determined as corrections of a general nature.

It will be seen that, whilst the principles and phenomena of
Aerodynamics can be illustrated qualitatively with ease in a modest
wind tunnel, the constant need for quantitative information makes
more serious demands and creates a study within itself.

in]

WIND-TUNNEL EXPERIMENT

51. Atmospheric Wind Tunnels — Open-Return Type

The cross-section of the experimental part of a wind-tunnel
stream may be square, round, elliptic, oval, octagonal, or of other
shape. The size of a tunnel is specified by the dimensions of this
cross-section. Apart from small high-speed tunnels actuated by a
pressure reservoir, the flow past the model is induced by a tractor
airscrew located downstream. The airscrew is made as large as
possible, if only to minimise noise, and its shaft is coupled direct to
the driving motor, the speed of which is controlled preferably by
the Ward-Leonard, Kramer, or similar electrical system. If C "is
the cross-sectional area and V the velocity of the experimental part
of the stream, the ' power factor ' P is usually defined as

__ 550 X input b.h.p.

But in some publications the reciprocal of this ratio is intended.

The term atmospheric applied to a wind tunnel means that the
density of its air stream is approximately the same as that of the
surrounding atmosphere. Some tunnels employ compressed or
rarified air, but they are few, and so the term is commonly omitted
in referring to the atmospheric class.

For some years many of the wind tunnels built were of the type
shown in Fig. 26, described as ' straight-through ' or ' open-return/

cas-

.-:£ \>

SCALE OP FEET

FIG. 26. — 4-FT. OPEN-RETURN WIND TUNNEL.

H, inlet honeycomb ; P, plane table ; S, guard grid ; D, regenerative cone ;
W, honeycomb wall.

Though the design has been superseded, numerous examples are
still in use. Air is drawn from the laboratory into a short straight
tunnel through a faired intake and wide honeycomb, the location of
the latter being adjusted to spread the flow evenly over the working
section. Subsequently the stream has most of its kinetic energy
reconverted into pressure energy in a divergent duct D, from

72 AERODYNAMICS [CH. Ill

which the airscrew exhausts the air into a ' distributor,1 a large
chamber enclosed by perforated walls W. The distributor returns
the air, with disturbances due to the airscrew much reduced, over a
wide area to the laboratory, which conveys it evenly arid slowly
back to the intake and thus forms an integral part of the circuit.
A consequent disadvantage is that the laboratory requires to be
reasonably clear of obstructions, symmetrically laid out, and also
large ; approximate dimensions for a tunnel of size x are : over-all
length, including diffuser, 14* ; height and width, 4£#. A second
disadvantage is lack of economy in running, the power factor P
having the high-value unity. In small sizes, however, the type is
simple to construct and convenient in use.

A boundary layer of sluggish air lines the tunnel walls, but away
from this Bernoulli's equation holds closely, showing a wide central
stream almost devoid of vorticity. This stream slightly narrows
along the tunnel owing to increasing thickness of the boundary layer.
Thus the streamlines are slightly convergent ; velocity increases
and pressure decreases along the parallel length. To compensate for
tnio characteristic variation, tunnels are sometimes made slightly
divergent.

The static pressure is obviously less within the tunnel than out-
side. At first sight it may appear feasible to calculate the velocity
at the working section from a measurement of the difference in static
pressure between there and some sheltered comer of the laboratory.
But losses in total energy occurring at the intake, principally through
the honeycomb, prevent this. The pressure in a pitot tube within
the working stream is less than the static pressure in the room. A
small hole is drilled through the side of the tunnel several feet up-
stream from the working section, and the pressure drop in a pipe
connected with it is calibrated against the appropriate mean reading
of a pitot-static tube traversed across the working section (excluding,
of course, the boundary layer). By this means velocities can after-
wards be gauged without the obstruction of a pitot-static tube in the
stream.

52. Closed-Return Tunnels

In the more modern tunnels of Fig. 27, the return flow is con-
veyed within divergent diffuser ducts to the mouth of a convergent
nozzle, which accelerates the air rapidly into the working section.
A ring of radial straighteners is fitted behind the airscrew to remove
spin and the circulating stream is guided round corners by cascades

(c)

FIG. 27. — RETURN-CIRCUIT WIND TUNNELS.
(a), enclosed section ; (b), lull-scale open jet ; (c), compact open jet ; (d), corner vane.

A.D.— 8*

74 AERODYNAMICS [CH.

of aerofoils or guide- vanes (see (d) in the figure for a suitable section),
which maintain a fairly even distribution of velocity over the
gradually expanding cross-section. The experimental part of the
stream is preferably enclosed, as at (a), but sometimes takes the
form of an open jet, as at (b) and (c). An open jet is distorted by
a model and is resorted to only when accessibility is at a premium.
These tunnels are often known as of ' closed-return ' or ' race-
course ' type. They effect a great economy in laboratory space,
only a small room being required round the working section, and
also in running costs, P having approximately the value J. Wood
is not a suitable material for construction, though often used,
because during a long run the air warms and produces cracks which
are destructive to efficient working since the ducts support a small
pressure.

A characteristic of prime importance is the contraction ratio of
the tunnel, defined as the ratio of the maximum cross-sectional area
attained by the stream to the cross-sectional area of the experimental
par*, A large contraction ratio effectively reduces turbulence but
increases the over-all length of a tunnel of given size, since divergent
ducts must expand slowly to prevent the return flow separating
from the walls. A rather long tunnel has the advantage of prevent-
ing disturbances from a high-drag model being propagated
completely round the circuit. Modern designs usually specify a
contraction ratio greater than 6 ; values for
the tunnels (a), (b), (c) in the figure are 6£, 5,
and 3|, respectively, (a) may be regarded as
suitable for general purposes, (b) illustrates
the full-scale tunnel at Langley Field, U.S.A.,
which has an oval jet 60 ft. by 30 ft. in
section, an over-all length of some 430 ft., and
a speed of 175 ft. per sec. with a power
input of 8,000 h.p. (c) indicates the maximum
possible compactness for this type of tunnel ;
developed at the R.A.E., it has been used
for sizes up to 24-ft. diameter.

» » 1 1 ' » i * i »

FIG, 28. — SPINNING TUN-
NEL.

M, flying model; O,
observation window ; N,
net for catching model ;
H, honeycomb.

53. Spinning Tunnel

A few vertical tunnels have been built,
as shown schematically in Fig. 28, for spinning
tests. An aeroplane may fly in a vertical
spiral with a velocity of descent VT, say. A

Ill] WIND-TUNNEL EXPERIMENT 75

question arising is whether operation of the aerodynamic control
surfaces will steer the craft into a normal flight path. To investigate
this, a light model of balsa wood, similar in disposition of mass as well
as in form, is set into corresponding spiral flight, a camera mechanism
operating the controls after a delay. Ignoring the effects of vis-
cosity, the Froude number V*/lg must be the same for craft and
model. If the latter is made to T^th scale, its velocity of descent
= JFF. This is a small speed, and it is feasible to employ a wide
vertical tunnel with an upwardly directed stream, so that the model
does not lose height and the action can be observed conveniently.
The difficulty with these tunnels is to prevent the model from
(a) flying into the wall, (b) spinning upwards or downwards. Accord-
ing to tests carried out on model tunnels, (a) can be overcome by
a suitable distribution of velocity along the radius, and (b) by
making the tunnel slightly divergent, which gives stability in
respect of vertical displacement, since the rising model then loses
flying speed, and vice versa.

54. Coefficients of Lift, Drag, and Moment *

In the general case of a body suspended in a wind tunnel Aero-
dynamic force is not a pure drag, but is inclined, often steeply, to the
direction of flow. This inclination is not constant for a given shape
and attitude of the body, but is a function of the Reynolds
number.

When the flow has a single plane of symmetry for all angles of
incidence of the body, the Aerodynamic force can be resolved into
two components in that plane, parallel and perpendicular to the
relative wind — the drag and lift, respectively. By Article 47 we
find for any particular shape and incidence a lift coefficient :

*CL=*L= -/.(*) . . . (74)

and a drag coefficient :

D - p

* There are two systems of coefficients in Aerodynamics. , In the now prevailing
system, associated with the symbol C, forces and moments are divided by the
product of the stagnation pressure for incompressible flow, viz. JpF* (cf. Art. 32),
and /* or /* ; in an earlier system, distinguished by the symbol k, the quantity pV*
takes the place of the stagnation pressure. Thus a ^-coefficient = J x the corres-
ponding C-coefficient, as indicated in (74), (76), and (77). Neither system has an
advantage over the other, but to secure a universal notation (^-coefficients have
superseded ^-coefficients in this country since 1937. They are generally adopted
in this book, but some matters are still expressed in the older system.

76 AERODYNAMICS [CH.

Most bodies tested are parts of aircraft, and L is then positive if it
supports weight when the aircraft is right way up. For any chosen
Reynolds number, we have

Aerodynamic force (4) = foV*PVCL* + CD«

and, if its inclination to the direction of lift
is y (Fig. 29),

tan Y = CD/CL . . (76)

AX/&D = CJCD is called the lift-drag ratio
and = LjD.

Without a plane of symmetry as above,
A will have a third component, called the
crosswind force.

Again assuming this plane of symmetry,
the line of action of A can be found from
its magnitude and direction and the
moment about some axis in the body
perpendicular to the plane, usually through
the quarter-chord point. This moment is
called the pitching moment Af. The
method of Article 47 gives for any parti-
cular shape and attitude a coefficient :

M - . . . (77)

FIG. 29.

M is positive when it tends to increase angle of incidence, i.e. to turn
the body clockwise in the figure.

Other moment coefficients will be introduced later when the
motion of aircraft is considered in greater detail. It should
carefully be noted that Q, CD," CM are different functions of R ;
we shall often omit a distinguishing suffix to / without implying
equality.

It has been stated that any agreed length may be adopted for I to
specify the size of a body of given shape and attitude. More
generally, any agreed area may be used for /*, or volume for I9.
Practice varies in the choice made. CL, CD are always calculated for
single wings on the area S projected on a plane containing the span
and central chord (line drawn from nose to tail of median section).
The length of the chord c is introduced as the additional length
required for CM (although not for other moment coefficients, when the
semi-span is used). Thus for wings :

CL « L/$PF«S, CD « D/*pF«S, CM =

Hi] WIND-TUNNEL EXPERIMENT 77

The parasitic, or ' extra-to-aerofoil/ drag of a complete aeroplane,
i.e. the drag of all parts other than the wings, may sometimes for
convenience be referred to 5. But usually for fuselages (aeroplane
bodies), struts, and the like, and sometimes for airship envelopes, /*
is specified by the maximum sectional area across the stream.
Another area frequently used for airship envelopes is (volume)2/8,
enabling the drags of different shapes to be compared on the basis of
equal static lift. It is seldom suitable to employ the same / to specify
both R and the coefficients ; for R, the length from nose to tail is
usually chosen.

55. Suspension of Models

It is evident that the foregoing and other coefficients can be
determined through a range of R by direct measurement, given suit-
able balances. These are grouped round the working section of the
tunnel, and the model is suspended from them. Their design and
arrangement are partly determined by the following consideration.

Suppose the true drag D of a model in a tunnel is required. Let
the suspension attachments (called, for short, the holder) have a
drag d when tested alone. Let the drag of holder and model be D'.
Except under special conditions we cannot write : D = Dr — d ;
the combination represents a new shape not simply related to either
part. The mutual effect of d on D, or vice versa, is termed the
mutual interference. An example is as follows. If a 6-in. diameter
model of an airship envelope be suspended by fine wires, and a
spindle, the size of a pencil, made to approach its side end-on, the
drag of the airship may increase as much as 20 per cent, before
contact occurs.

The approximation used in general depends upon the interference
being local. A second holder is attached to a different part of the
model and a test made with both holders in place. Removing the
original holder and testing again with only the second holder fitted
gives a difference which is applied as holder correction to a third test
in which the original holder alone is present. The approximation
gives good results, provided neither holder creates much disturbance,
to ensure which fine wires or thin streamline struts are used.

Fig. 30 shows as a simple illustration an arrangement suitable for a
heavy long body having small drag. Near the nose the body is
suspended by a wire from the tunnel roof, while a ' sting ' screwed
into the tail is pivoted in the end of a streamline balance arm, for the
most part protected from the wind by a guard tube. If the guard

AERODYNAMICS

[CH.

tube is of sufficient size to deflect the stream appreciably, a dummy
is fixed above in an inverted position. Sensitivity, in spite of the
heavy weight of the body, is achieved by calculating the fore-and-
aft location of the wire to make, following small horizontal displace-
ment, the horizontal component of its tension only just overcome
that of the compression in the balance arm. To find the effective

FIG. 30. — TESTING A HEAVY MODEL OF Low DRAG.
G, guard tube ; P, scale pan ; S, sting ; T, turnbuckle ; W, cross-hair.

drag of the wire, another test is made with a second wire hung from
the nose as shown at (a) and attached to the floor of the tunnel.
Next, the sting is separated slightly from the balance arm, support
being by the wires (6) from the roof, and the effective drag of the
balance arm measured with the body almost in place. Finally, the
model can be suspended altogether differently, from a lift-drag
balance as at (c), the wires and original balance arm being removed,
and the small effective drag of the sting estimated by testing with it

Ill] WIND-TUNNEL EXPERIMENT 79

in place and away. At the same time special experiments can be
made to investigate the interference, neglected above, between the
sting and the original balance arm. It will be appreciated that the
reason why the arrangement (c) is avoided except for corrections is
that the spindle, although of streamline section, would split the
delicate flow near the body, and artificially increase its drag. The
model fuselage shown may have a small lift. To prevent consequent
error in drag measurement, the wire and balance arm must be
accurately vertical for a horizontal wind. This is verified by hanging
a weight on the body without the wind, when no drag should be
registered.

56. The Lift-drag Balance

V-

When several force and couple components act on a model it is
desirable for accuracy to measure as many as possible without dis-
turbing the setting of the model. Omnibus balances designed for this
purpose tend to be complicated, and reference must be made to
original descriptions. An indispensable part of the equipment of a
tunnel, however, is an Aerodynamic balance that will measure lift
and drag simultaneouslyjand preferably at least one moment at the
same time.

Aerofoil Kyn^ Diaphragm v

Tunnel Wall — %''W

FIG. 31. — SIMPLE LIFT-DRAG BALANCE.

A simple form of lift-drag balance is illustrated in Fig. 31. The
main beam passes through a bearing B centrally fixed to a hard
copper diaphragm, 5 in. diameter and 0-003 in. thick, clamped to a
flange of a casting which abuts on a side wall of the tunnel through
soft packing to absorb vibration. The diaphragm gives elastically,
permitting the beam to deflect in any direction almost freely between
the fine limits imposed by the annular stop O which is opened by the

80 AERODYNAMICS [CH.

lever T while observations are being taken. The diaphragm suspen-
sion prevents leak into the enclosed-type tunnel assumed ; it may be
replaced, if desired, by a gymbals with an open-jet tunnel, but sensi-
tivity is then more difficult to maintain with large forces. The
sensitivity of the balance described is 0-0003 Ib. The bearing
permits of turning the beam about its axis quickly and accurately
by means of the worm gear W, an angular adjustment that is often
useful, e.g. when testing an aerofoil of the form which can be sus-
pended by screwing a spindle into a wing-tip as shown in the figure.
Lift is measured by adjusting a lift rider on the main beam and by
weights on the scale pan L. The free end of the main beam carries a
knife-wheel E, engaging a hardened and ground plate at one end of a
horizontal bell-crank lever, of sufficient leverage to ensure that the
ainaH end movement of the main beam is negligible. This lever is
mounted on vertical knife edges, and transmits drag to a subsidiary
balance, with a drag rider and scale pan D.

Horizontal lift-drag balances are simple to construct and also
particularly convenient for testing square-ended aerofoils, negligible
interference occurring between the aerofoil and a spindle screwed
into its tip. They are inconvenient for aerofoils having thin tips
and are not readily adaptable to measure pitching moments. Their
usefulness is enlarged in combination with a simple steelyard
mounted on the roof of the tunnel, as described in the next article.
But experience with this double-balance method of testing suggested
the more adaptable modern types of balance described in principle
later.

57. Double Balance Method of Testing an Aerofoil

The distinguishing feature of a good aerofoil, or model wing, at
fairly large Reynolds numbers is that its Aerodynamic force A is, at
small angles of incidence, nearly perpendicular to the stream ; LjD
may then be 25 and y of (76) 2-3°. The point P (Fig. 32), at which A
intersects the chord, of length c, is called the centre of pressure and
NP/c the centre of pressure coefficient &CP. The method described
enables L, D and P and consequently M to be determined with only a
simple roof balance and a lift-drag balance. The aerofoil is suspended
from the former by wires attached to sunk eye-screws at W and from
the latter through a sting pivoted at E. A drum carried by the
roof balance enables the length of the wires to be adjusted and hence
the incidence a. The model is suspended upside down to avoid the
use of a heavy counterpoise, although a small one is desirable with a

WIND-TUNNEL EXPERIMENT

HI]

light model for safety, to keep the wires taut, and to permit measure-
ment of small upward forces — negative lifts.

Part L' of the lift L is taken at W, the remainder U at E. The
wires are set truly vertical at some small incidence a, when they will
be also vertical at a small negative a, but at no other incidence. Let

FIG. 32.

6 be their small inclination to the vertical, the stream being assumed
truly horizontal, and T that part of their tension due to A. They
support a part T sin 0 of the drag D, only the remaining part d being
supported at E. The lift-drag balance connected to E provides the
only means of measuring D. Thus 0 must be corrected for accur-
ately and the method adopted is as follows. At any setting of the
aerofoil the zeros of the lift-drag balance are observed, before starting
the wind, with and without a known weight hooked on the model.
An apparent drag is thus found for a known value of T at the parti-
cular value of 0 corresponding to a, but which need not be known.
A proportionate correction appropriate to the value of T measured
when the wind is on can then be applied to drag observed at E. This
correction requires to be determined for all values of a.

Measurements of drag must further be corrected for (a) part of
the drag of the wires, for which purpose the measurements may be
repeated with additional wires attached in a similar manner, or a
calculation may be made based on Fig. 25, the geometry of the rig
and the thickness of the tunnel boundary layer ; (6) the effective

82 AERODYNAMICS [CH.

drag of the lift-drag balance 'arm, determined as in Article 55, a
being varied through the complete range studied ; (c) the effective
drag of the sting, obtained by measuring drag with and without the
sting in place at all incidences with the model suspended in some other
manner, e.g. by a spindle fastened to a wing-tip.

Referring to Fig. 32, L = L' + L* , and taking moments about E
we have

U . I cos 0 = A . a = Tl cos (p - 6)
giving

L9 =7(1 + 0 tan (3)

since 6 is small. Also D = TQ + d

Tl
a = -j(cos (3 + 6 sin p)

where A = V(if + &)

and y = tan ~l (D/L).

Finally NP = c — {a sec (a — y) — s}-

58. Aerodynamic Balances

The foregoing method is simplified by fixing W and adjusting a
by displacing E ; the front wires may then form two longitudinal
vees, and a vertical sting wire at E replace the lift-drag balance arm.
The whole of the drag, as well as the major part of the lift, is taken
by the vee-wires, and the sting wire supports only the remainder
of the lift.

This in brief is the principle of the Farren balance, shown
schematically at (a) in Fig. 32A. Part of the lift and the entire
drag are communicated by two parallel pairs of vee-wires, inter-
secting at W, to the frame F located above the tunnel and pivoted
vertically above W. The drag is transmitted by an increase of
tension in the front wires of the vees and a decrease of tension in
the back wires, and thus a counterpoise must be suspended from
a light model of high drag in order to keep the back wires taut.
Such a counterpoise is advisable in any case as a safeguard, and
then care need not be taken to locate W well in front of the centre-
of-pressure. The frame is weighed in the balances L and D for the
lift and drag communicated to it. The sting wire, shown fastened
at E to the fuselage of a complete model in the figure, remains
truly vertical with change of incidence by virtue of being raised or
lowered by a stirrup R which is parallel to EW and pivoted vertically
above W. The familiar problem is to measure the remaining part

Ill] WIND-TUNNEL EXPERIMENT

of the lift supported by
the sting wire without
interfering with the drag
balance. This is achieved
by pivoting the bell-
crank lever, which sup-
ports the stirrup, level
with the pivot of the
frame F. Thus these two
pivot lines are coincident,
although in the figure
they are shown slightly
displaced from one an-
other for clearness. If
the pivot of the bell-
crank lever is carried on
the lift beam, the whole
of the lift is transmitted
to that beam, and the
balance marked M is used
only to determine the
pitching moment of the
Aerodynamic force about
W.

The balance shown at
(ft) in the figure makes
use of a different system,
enabling all pivots to be
located outside the tunnel.
The model is suspended
from the platform F by
any convenient means
and, provided the two
lift beams shown are of
equal length, the true
lift and drag are measured
whatever the position of
the model relative to F.
However, the pitching
moment is determined
about the line joining
the intersections of the

(a)

(b)

w/////

FIG.— -32A. AERODYNAMIC BALANCES

84 AERODYNAMICS [CH.

centre-lines produced of the two inclined pairs of sloping struts
which support F, i.e. about W in the figure. This is readily verified
by considering the effect of a load acting in any direction through
W ; it would evidently cause tensions and compressions in the
sloping struts but no force in the moment linkage. Hence the
suspension from the platform will in practice usually be so arranged
that the pitching moment is measured about a significant point in
the model. The linkage connecting the drag and moment balances
should ensure that these give the drag and moment separately, i.e.
without interfering with one another.

The third balance (c) is an inverted form of (b) with other
modifications. The moment balance is mounted on the lift plat-
form G instead of being attached to a fixed point, a step which
eliminates the necessity for a linkage to prevent interference
between the moment and drag measurements. All weights used
on the moment balance are stored on the lift platform so that their
adjustment will not affect the lift reading. The drag frame H is
supported in a parallel linkage so that fore and aft movements can
occur without vertical displacement, and in consequence excessive
static stability is avoided without the use of counterpoises.

The foregoing illustrates only a few of the many devices put to
use in the design of a modern Aerodynamic balance. For clearness,
the three balances have been described in 3-component form, but
all are readily adaptable to cope with additional components.
The following constructional features may also be noted. Elastic
pivots are preferred to knife-edges or conical points and commonly
take the form of two crossed strips of clock-spring. The amount
of damping required is extremely variable, and therefore the electro-
magnetic method is preferred to a plunger working in oil. When
a balance is inaccessible or there is need to save time in operation,
weighing and recording can be carried out mechanically.

59. Given tunnel determinations of lift, drag, etc., freed from
parasitic effects, various corrections are necessary before they can
be applied to free air conditions at the same Reynolds number.
These are in respect of : (1) choking of the stream by a body of rela-
tively considerable dimensions, (2) deviation of the undisturbed
stream from the perpendicular to the direction in which lift is
measured, (3) variation of static pressure in the undisturbed stream,
(4) effects of the limited lateral extent of the stream, applying
principally to wings, and developed in Chapter VIII. A further
cause of difference is introduced in Article 65.

(1) It is possible to express the argument of Article 35 in approxi-

WIND-TUNNEL EXPERIMENT

III]

mate numerical form for a given shape, when it is seen to follow that
the choke correction is small. For a body whose diameter is J that
of the tunnel the correction is usually < 1 per cent.

(2) Let the stream be inclined downward from the horizontal at
a small angle p, and, taking the familiar case of an aerofoil upside
down, let its aerodynamic force be A and its true lift and drag L and
D, respectively. The apparent lift and
drag measured, however, are La and Da
(Fig. 33), We have, assuming p small —

L = A cos y D = A sin y

La = A cos (Y - P)
Da = A sin (Y - p)

= A (sin Y — P cos y)

= D - pi.

Thus the error in La is negligible, but
this may be far from true of Daf for we
have

(78)

FIG. 33.

Upward inclination of the stream leads
to an error in drag of the same magnitude but opposite in sign.

Example : If p == £°, and L/D ==>20, the error in D is ± 17 J per
cent.

This error can be removed by testing the model right way up and
upside down, and taking the mean. The process is laborious, how-
ever, and a correction factor for general use is worked out by an
initial test of this kind. Where their design permits, balances are
carefully set on installation so as to eliminate the error as far as
possible.

(3) Convergence or divergence of the stream leads to an error due
to the pressure gradient that exists in the direction of flow prior to
introducing the model. In the former, the more usual case, pressure
decreases downstream (x increasing). Owing to the short length of
the model dp/dx may be assumed constant, and to this approxima-
tion is easily determined experimentally. The maximum con-
vergence in a parallel-walled tunnel is only about J°.

Complete analysis of the problem presents difficulty, but an
inferior limit to the correction is readily calculated by a method that
will now be familiar. Considering an element cylinder of the body,
of cross-section AS and length /, parallel to the direction of flow and

86 . AERODYNAMICS [CH.

coming to ends on the surface of the body, the downstream force on
it, if we apply a method analogous to that of Article 8, is readily found
to be — (dp/dx) (AS . /). The whole volume V can be made up of
such cylinders, giving for the downstream force on the model
— (dp/dx)V, which is essentially positive for convergence. This
force has nothing to do with drag, vanishing when the stream is
parallel or the model moves through free air, and measurements
must be decreased on its account. The correction is important for
low resistance shapes such as airship envelopes and good aeroplane
bodies and wings at high-speed attitudes.

Further analysis shows that the volume should be greater than
that of the body, an increase of 5-10 per cent, being required for
long bodies of revolution, 10-15 per cent, for wings, and 30 per cent,
for compact strut shapes, approximately. The correction does not
vanish in the case of bluff shapes of small volume, but it is then
numerically unimportant. (See also Article 230B.)

59A. Pitot Traverse Method

The drag of a two-dimensional aerofoil can be estimated from an
exploration of the loss of pitot head through a transverse section of
its wake. This loss will be denoted by a non-dimensional co-
efficient h, as follows. Let U, pQ be the undisturbed velocity and
pressure, respectively, and q, p the corresponding quantities at any
point in the wake. Then the loss of pitot head at the point is

P* + *pt/« - (P + *P?a) = h . iplT*.

It is much more marked close behind the aerofoil than farther down-
stream, where the wake has diffused outward.

Consider first a section of the wake sufficiently far behind the
aerofoil for the pressure to be equal to pQ and the velocity to have
become parallel again to the relative motion, a state distinguished
by writing q = u. Through an element 8y of this section, of unit
length parallel to the span of the aerofoil, the mass passing in unit
time is p«8y, and the rate of loss of momentum parallel to the
relative motion is pwSy . (U — u). Hence the drag Z)0 of unit
length of the aerofoil is given by

Dg = p/«(C7 — u)dy
or

Ill] WIND-TUNNEL EXPERIMENT

But u/U = (1 - A)*. Hence

Far behind the aerofoil h will be small and the term in the square
brackets can be expanded as follows —

1_(1- tA-i*i + ...) = t*.
approximately, so that in this case

Defining a drag coefficient C^Q as equal to DQlfoU*c, where c is
the aerofoil chord, the result can be written

CDO =

(iii)

This coefficient is known as the profile drag coefficient of the aerofoil.
It includes the entire drag under two-dimensional conditions but
only part of the drag under three-dimensional conditions, except at
the incidence for zero lift ; at other incidences a wing has in addition
an inditced drag coefficient, arising from the continuous generation
of lift Aerodynamically and appearing as a modification of the

pressure distribution for two-dimensional flow. The pitot traverse
method finds uses in the wind tunnel, where two-dimensional flow
can be simulated, but its chief application is to flight experiments.
Exploration on the above lines of the wake of a wing can give only
its profile drag, but its induced drag can be estimated separately
by calculation, as will be found in Chapter VIII. A difficulty
arising in flight is that the pitot traverse must be made close behind
the wing, so that the pressure differs from pQ and a correction to
(iii) becomes necessary. The experimental section near the wing
will be distinguished by suffix 1, see Fig. 33A.

This correction is rather uncertain. Jones* has suggested ignor-
ing the turbulence in the wake and relating the pressure and velocity

* Jones (Sir Melvill), A.R.C.R. & M. No. 1688, 1936.

88 AERODYNAMICS [CH.

at section 1 to those at the distant ^section, where p «= pQ, by
Bernoulli's equation, applied along a suppositions mean stream-
tube. Then :

Pi + iPft1 - Po +

Writing

* -

this gives

Again,

- (A

so that

| = (1 - h, - ktf ... (V)

Let w denote distance perpendicular to the direction of mean
motion at section 1. Then for incompressible flow q£n = «Jy and
(i) becomes

s - (^(i-u
&- }~UV U

Substituting from (iv) and (v) and again introducing the drag
coefficient,

1 - (1 - AJ*J rfn . (79)

This result is known as Jones1 formula. Tested in a full-scale
wind tunnel, it was found* to be accurate within experimental errors
along the middle three-quarters of the span of a rectangular wing.
Nearer the wing-tips the induced flow associated with the produc-
tion of lift under three-dimensional conditions makes the method
inapplicable. Restrictions of another kind have been discussed
by Taylor, f

A different treatment of the problem has been given by Betz.f
His formula includes provision for dealing with the induced flow
caused by three-dimensional production of lift.

* Goett, N.A.C.A. Report No. 660, 1939.
t Taylor (Sir Geoffrey), A.R.C.R. & M. No. 1808, 1937.

j Betz, Z.F.M., vol. 16, 1925 ; see Arts. 79-81 by Prandtl in Tietjens, ' Applied
Hydro- and Aero-Mechanics/

Ill] WIND-TUNNEL EXPERIMENT 89

The pitot traverse method of drag measurement offers such
manifold advantages that the subject is an old one and has received
attention on many occasions. The exact theory is complicated,
however, and formulae obtained by simple means require to be
established by experiment, The method can be relied upon to give
a close estimate of drag under fairly favourable conditions ; viz.
briefly when the pressure in the wake differs little from pQ and the
velocity trough is rather shallow. These conditions imply, especially
if CDO has a considerable value, that the traverse should be made
well downstream, but this is obviously inconvenient in flight, whilst
in tunnel experiment it may sometimes vitiate the two-dimensional
assumption. Again, the section behind which a traverse is made
may not be truly representative of the average section of a wing or
aerofoil. Such difficulties partly explain discrepancies that are
found to exist.

The exploring pitot tube should be fine in order to avoid a system-
atic experimental error. The^ effect of compressibility on the
method has also been examined.* The estimates are not affected
by any pressure gradient that may exist in the empty tunnel.

60. An example of wing characteristics obtainable by the method
of Article 57 at small scale, e.g. in a 6-ft. open-jet tunnel at 100 ft.
per sec., is given in Fig. 34, corrections noted in Article 59 having
been made so that the results apply to free air conditions at a
small Reynolds number. The aerofoil is of the section shown,
known as Clark YH, with a ratio of span to chord, called aspect
ratio, of 6.

Features fairly typical of aerofoils in general may be noted.

Zero lift occurs at a negative incidence, usually small. The value
— 3° shown in the present case is arbitrary, in the sense that it
depends upon how a is measured. The present aerofoil has a partly
flat lower surface, which is used to define inclination to the wind.
Another aerofoil might have a slight concavity in the lower surface,
when the common tangent would be employed. Most aerofoils are
bi-convex, however, when the line joining the centres of curvature
of the extreme nose and tail defines a.

Lift attains a maximum at a moderate angle, 15° in the present
instance, after which it falls. This incidence is known as the critical
or stalling angle and, combined with the maximum value attained
by CL, is of importance in connection with slow flying. The open-
jet tunnel determines this feature more reliably f than the enclosed-

* Young, A.R.C.R. & M. No. 1881, 1938.

f Bradfield, dark, and Fairtfcorne, A.R.C.R. A M., 1363, 1930.

075

0° 5° 10* 15*

INCIDENCE, <x

20°

FIG, 34.— CHARACTERISTICS OF CLARK YH WING (ASPECT RATIO 6) AT SMALL
SCALE (6-rr. OPEN-JET TUNNEL AT 100 FT, PER SEC.).

WIND-TUNNEL EXPERIMENT

CH. in]

section tunnel, which tends to flatter compared with free air. The
flow is often delicate in this region, some lift curves branching, and
different coefficients being obtained according as to whether a is
increasing or decreasing.

Minimum drag occurs when the lift is small but maximum LfD at
a considerably larger incidence. Drag and the angle y begin to
increase rapidly at the critical angle.

At — 1-3° the centre of pressure is midway along the chord. It
moves forward as incidence is increased up to the critical angle, and
then back. This travel results from striking changes which occur
in the shape of the pressure diagram, illustrated for a rather similar
aerofoil in Fig. 35. Between — 2-7° and — 4-5°, approximately,

D5r

-O5
-1-0
-15

-20

OUO°

10C

FIG. 35. — LIFT PRESSURE DIAGRAMS FOR THE MEDIAN SECTION OF AN AEROFOIL
(BROKEN LINE APPLIES TO LOWER SURFACE).

the C.P. is off the aerofoil. CCP = ± oo at a = — 3°, meaning that
when the resultant force is a pure drag, lift being zero, there is a
couple on the aerofoil ; the two loops seen in the pressure diagram
for 0° become so modified at — 3° as to enclose equal areas. Thus
the C.P. curve has two branches asymptotic to the broken line in
Fig. 34 ; part of the negative lift branch is shown near the left-hand
zero of the scales.

The travel of the C.P. for a < 16° indicates a form of instability.
To see this, imagine the aerofoil to be pivoted in the tunnel about a
line parallel to the span and distant 0-3 chord from the leading
edge, and to be so weighted that it is in neutral equilibrium for
all angles without the wind (an experiment on these lines is easy
to arrange). If now the model be held lightly at oc = 5°, the
incidence at which the C.P. is cut by the pivot line, and the wind
started, a couple tending to increase or decrease a will instantly
be felt, small disturbance of a displacing the C.P. in such a
direction as to increase the disturbance. The aerofoil will ride in
stable equilibrium, however, at a = 20°. It would also be stable

AERODYNAMICS

[CH.

if pivoted in front of 0*25 chord, but this case is only of interest
in connection with the auxiliary control surfaces of aircraft. The
C.P. curve is physically indefinite, in so far as it would have a
different shape if we defined the C.P. as the intersection by A of
some line parallel to the chord but displaced from it. Thus in
the above experiment different results would be obtained if the
pivot line were displaced from the chord plane.
Fig. 36 contains the essential information of Fig. 34 plotted in

more compact and practical
form, CL being more generally
useful than a as the indepen-
dent variable. The moment
coefficient given defines the
pitching moment about a
line one-quarter chord behind
the leading edge of the aero-
foil, often preferred for greater
precision to that of Article
54. Its middle point is called
the Aerodynamic centre.

61. Application of Complete
Model Data

Where f(R) has been found
in the tunnel through a
sufficient range, calculations
may be made for the shape
of body concerned in a
variety of practical circum-
stances, as illustrated in the
two examples following.

(a) A temporary wire,
J in. diameter and 2 ft. long,
is fixed parallel to the span
of a wing just outside its
boundary layer, above * a
position where the pressure

drop amounts to 15*36 Ib. per sq. ft. when the aeroplane is flying at

100 m.p.h. at low altitude. Find the drag of the wire under these

conditions.
First determine the relative velocity of the wire. Writing p, V

for the local pressure and velocity (ft. per second) where it is ex-

-0-4

-002

-0-04

-006

FIG. 36. — CHARACTERISTICS OF CLARK YH
WING (ASPECT RATIO 6) AT THE SMALL
SCALE OF FIG. 34.

CM «= pitching moment coefficient about a
line J chord behind the leading edge.

in] WIND-TUNNEL EXPERIMENT 93

posed, and distinguishing normal values by suffix 0, by Bernoulli's
theorem ^^ _ ^ ^ ^ _ ^ ^ ig<36 ft ^ ft^

giving V =s 185-5 ft. per sec.

Now / = diameter of wire = TV ft., v = 1-56 x 10~ * sq. ft. per
sec. Hence :

giving, from Fig. 25, CD = M6

0-58 X 0-00238(185-5)' X 2

Drag ==

= 0-99 Ib.

(b) The lift coefficient of a wing of the section shown (known as
R.A.F. 48) and span/chord ratio (aspect ratio) = 6, set at 15°
incidence, is found in the wind tunnel to vary as in Fig. 37, through

FIG, 37, — APPROXIMATE SCALE EFFECT ON LIFT COEFFICIENT OF R.A.F. 48
WING AT 15° INCIDENCE.

the range of R given. A parasol monoplane fitted with a wing of
this shape, chord = 6 ft., is required to approach a landing field
situated at 6000 ft. altitude at the same incidence and 60 m.p.h.
A.S.I, (indicated air speed). What lift will the wing exert when
standard atmospheric conditions prevail ?

94 AERODYNAMICS [CH.

From Table III, Article 14, temperature is 6'1° C. and relative
density 0*862.

p \ 273 / 0-00238 X 0-862

= 1*77 x 10 ~4 sq. ft. per sec.
(cf. Article 25). 60 m.p.h. = 88 ft. per sec. and the true velocity

V = , = 94-8 ft. per sec.

Vo-862

_ 94-8 X 6 X 10*

R = - — - = 3-21 X 10*.

Hence, from the figure CL = 1-248, and, since wing area =
(6 X 6)6 = 216 sq. ft.,

L = 0-624 X 0-00238(88)* X 216 = 2484 Ib.
or 11-5 Ib. per sq. ft.

The following may also be verified. A scale model of 1 ft. chord
would have, at 60 m.p.h. at sea-level, a Reynolds number =
0-65 X 106. Its CL would be 1-164, and it would lift 10-63 Ib. per
sq. ft. or a total of 63-8 Ib. The same fairly low Reynolds number
would apply to the full-scale wing held in a natural wind of 10
m.p.h., when the total lift would be the same and its mean intensity
0-296 Ib. per sq. ft.

62. Arrangement of Single Drag Experiment

Such complete data as in the last article are rare. Frequently the
drag of some aircraft part is desired accurately only under particular
conditions, e.g. at top speed at a certain altitude. From these
specifications and the size of the part the full-scale Reynolds number
can be calculated, and sometimes a single decisive test arranged in
the wind tunnel under dynamically similar conditions.

Examples : the drags are required of the following aircraft parts
exposed at A.S.I. = 150 m.p.h. at 10,000 ft. altitude : (a) a stream-
line static balance weight of 2 in. diameter, (b) a long strut whose
streamline section is 6 in. in length. Arrange suitable experiments
in a 4-ft. wind tunnel at sea-level working at 50 ft. per sec.

The true relative velocity of the craft is, from Article 33,

150 88

X — « 256 ft. per sec.

VO-738 60

Assuming standard atmospheric conditions, the temperature =
— 4-8° C. and, as in Article 61 (6), v is found to be 2-01 x 10 ~ *. For
the tunnel 15° C. may be assumed, so that v = 1-56 x 10 -«.

WIND-TUNNEL EXPERIMENT

ni]

Models geometrically similar to the parts will be tested in the
tunnel. Distinguishing experiment by suffix T and full scale by
suffix F, for dynamical similarity we have JRT = J?F.

(a) Let d be the diameter of the model of the balance weight.
Then

d X 50 (1/6) X 256

1-56

2-01

d = 0-662 ft. = 8 in.

The drag coefficient measured on a model 4 times as large as the
actual weight will apply exactly to full scale under the prescribed
conditions. From (72a) the forces on the model and the weight will
be in the ratio

0-738 \2-01

= 0-816.

(b) Similarly the tunnel model of the strut should be larger than
full scale in the ratio 4. Applying the factor to the strut section
gives a model section of length 4 x 6 = 24 in. Since only a
short axial length can be accommodated in the tunnel, the test is
arranged under two-dimensional conditions, i.e. experiment is
limited to finding the drag per ft. run of the strut well away from its
ends. Fig. 38 shows the rig. A model of axial length 2 ft. 6 in.,
say, swings with
small clearance be-
tween shoulders of
the same section
fixed to the tunnel
walls. These dummy
ends separate the
model from the walls,
eliminating error due
to the tunnel bound-
ary layer. Suspen-
sion may be by wires
and sting, or by a

FIG. 38. — ARRANGEMENT FOR TESTING UNDER
APPROXIMATELY TWO-DIMENSIONAL CONDITIONS.

M, model ; E, E, shoulders or dummy ends fixed to
wind-tunnel walls.

spindle passing

through one of the shoulders, which will then act as a guard tube.

The drag coefficient determined will apply exactly to the full-scale
strut at the speed and altitude given, except near its sockets. The
force measured will be simply related to that on a length 2 ft. 6 in. -f-
4 = 0*625 ft. run of the actual strut.

AERODYNAMICS

[CH.

63. Compressed-air Tunnel

The foregoing examples illustrate that, to secure dynamical
similarity, models will not as a rule be smaller than full-scale parts.
The restriction is unimportant in the case of small components, but
destructive for large parts, such as wings. An element of the span of
the full-scale wing of a small aeroplane can be tested under two-
dimensional conditions in a 4-5-ft. tunnel as described above for a
strut model, provided incidence is closely that for zero lift. It will be
found later on that Aerofoil Theory and Skin Friction Analysis can
then be used to deduce the drag of the wing in free flight through
a useful range of incidence. But certain important phenomena
occurring at considerable incidences must be measured directly on
models of complete wings. Not more than 8-in. chord could then be
used in a 5-ft. tunnel. Thus a model wing is seldom larger than £th
and may be smaller than -^th scale. To realise full-scale Reynolds
numbers by testing at speeds six to twenty times as great as those of
flight is impracticable both on economic grounds and as vitiating the
incompressible flow assumption.

In some national tunnels, however, the air circulates in a com-
pressed state, pressures of 25 atmospheres being reached. By
Maxwell's Law (Article 25) jx is independent of density at constant
temperature, when v oc 1/p and R oc p. Hence for R constant

VF ~~ ^ VF h' Pr'

where suffix T refers to the model and F to full scale. If, for
example, the last factor is ^V and ^T/^F = iV» Vr/VF = f .

Fig. 39 illustrates the compressed air tunnel at the N.P.L. It is
of annular return flow type, the working jet being 6 ft. diameter and
the enclosing steel shell 18 ft. diameter and 2| in. thick in its
cylindrical part. A 450-h.p. motor gives a wind velocity of 90 ft.
per sec. through the working section. After rigging a model, the

tunnel is sealed and pumped

H tsi up by a large compressor

plant in an adjoining room.
Forces and moments are
measured by special balances
located within the shell and
controlled electrically.* The
exhaust from the tunnel, after
the test, is utilised to drive small
high-speed tunnels.

FIG. 39. — COMPRESSED-AIR TUNNEL.

N, N, radial vanes to prevent swinging
of flow.

* Relf, Jour. Roy. Aero. Soc.t Jan. 1936.

HI]

WIND-TUNNEL EXPERIMENT

If 6 is the compression ratio required in the C.A.T. to secure a full-
scale low-altitude value of R, so that VI for the model is 1/6 times
that for full scale, then by (72) the ratio of any component force on
the model to the corresponding full-scale component is also 1/6.
For example, with 6 = 25 the aerofoil lift (Z,T) for a wing of 5 tons
lift (IF) would be 448 Ib. If the geometrical scale of the model
were TaTth we should have for the ratio of the mean intensities
of lift

IT//T* _ 16a

Z,F//F«~~l!5

or the loading on the model would be 10 times that on the full-scale
wing, and might reach 4 cwt. per sq. ft. Such intense forces readily
distort C.A.T. models, which are therefore often shaped from metal
castings.

64, Practical Aspect of Aerodynamic Scale

Table V relates to aeroplane wings and complete models of wings,
and indicates the maximum Reynolds numbers obtained in various
types of wind tunnel and the range of Reynolds numbers char-
acterising various aircraft categories. R is specified on the mean
chord. Maximum sizes are assumed for the models and involve
important corrections for the limited width of the tunnel stream
(Chapter VIII).

TABLE V
REYNOLDS NUMBERS OF WINGS

Speed
(ft. per sec.)

R -r 10«

Tunnel Experiment :
12-36 h.p. 4 ft. enclosed-section (A)

100

600 h.p. 6-6 ft. open-jet (A) .
12,000 h.p. 60 X 30 ft. open-jet (A)

300
200

U
0*

600 h.p. 6 ft. jet (25 atmospheres)

100

10J

9.000 h.p. 13 X 9 ft. enclosed-section (4 atmo-

spheres)

300

Full Scale Flight :

Small and light aeroplanes
Small fast aeroplanes .
Medium-size aeroplanes
Medium-size (stratospheric)
Specially large aeroplanes

80-250
100-600
110-450
700
130-400

2J-7
4-25
9-36
16
15-46

(A) denotes atmospheric pressure.

In compiling this list, which is by no means exhaustive, some
known extreme cases have been omitted in order to preserve a

A.D.— 4

98 AERODYNAMICS [CH.

generally representative view of the position. The tunnels are not
confined to existing plant ; that last mentioned is designed primarily
for high Mach numbers (as will be described later) but can be used
as a compressed-air tunnel, as shown. The comparatively small
Reynolds number typical of stratospheric flight is due to the large
value of v at 40,000 ft. altitude ; in regard to flight at this altitude,
an atmospheric tunnel has a compression ratio of 4. All the other
aircraft data relate to comparatively low altitudes.

It will be seen that a small tunnel using highly compressed air is
much the most economical for a straightforward test on an aerofoil
when the Mach number can be ignored. The 6-ft. size with pressures
up to 25 atmospheres covers the entire range of Reynolds number
for small aeroplanes of low power, and the landing conditions and
stratospheric Reynolds numbers of all but the largest aeroplanes,
for experimental Reynolds numbers can be increased to about 25
million for small incidences by testing under two-dimensional
conditions and applying a theoretical correction for the change to
three dimensions, checked experimentally in the same tunnel at a
smaller scale. A very large tunnel is desirable in other connections,
e.g. when access is required to the stream during a test, or details
are concerned which cannot be reproduced in small models ;
instances of such details are engine cowlings and Aerodynamic
controls ; but it is often claimed that these purposes can be served
without going to the extreme of a full-scale tunnel. The relative
advantages of the two methods have, indeed, long been contended.
For very high compressions, models are expensive to construct and
the exceedingly heavy air creates experimental difficulties, in
connection with deflections, that are sometimes serious ; there is a
case for restricting the compression to less than 8 atmospheres, and
increasing the size, though the advantage in respect of power is then
greatly reduced. On the other hand, an aeroplane cannot in general
be tested in a full-scale tunnel, under conditions quite different from
those of flight, without special strengthening, a process which is not
simple to carry out, Further reference to the matter will be made
in the chapter on testing at high speeds. Meanwhile the conclusion
will be drawn that wind tunnels, of whatever kind, capable of realising
full-scale Reynolds numbers are costly, and that experiments will
usually be made, therefore, at much smaller Reynolds numbers.

65. Scale Effects (a)

Since tunnel measurements are made in general at too low a
Reynolds number, important differences are to be expected on the

WIND-TUNNEL EXPERIMENT

Ill]

craft. These are termed scale effects. By ' scale ' is meant, of
course, Aerodynamic scale. But the practical term has two
meanings, of which we now consider the first, i.e. the total change in
f(R) through the interval of R between experiment and flight.

Rapid changes often occur in a 5-ft. atmospheric tunnel of moder-
ate speed, e.g. the dip in the CM curve of Fig. 36 quickly disappears
on increasing scale. Thus it is advisable to test at as large a scale as
possible, however far this may be from that of flight. Certain

0-6
0-6

0-4
0-2

0-08

0-06

0008j

FIG. 40. — EXAMPLES OF SCALE EFFECT.

(a) Spheres. Full line : smooth flow ; dotted line : turbulent wind tunnel ;
hatched area : steadier tunnels.

(b) Streamline strut of fineness ratio 3. Full line : CD reckoned on maximum
sectional area across the stream ; dotted line : CD reckoned on maximum sectional
area parallel to the stream.

(c) Smooth tangential boards or plates. Full line : smooth flow ; dotted line :
turbulent tunnel ; chain line : some experiments under fairly steady conditions.

subsequent changes can be estimated from theory, as we shall find
later on, but others, of which Fig. 37 is an example, rest entirely on
experiment or engineering experience.

Fig. 40 gives further examples off(R). CD is reckoned for (a) and
the upper full-line curve of (b) on maximum sectional area across the
stream ; for (c) on maximum area projected perpendicular to the
stream. To facilitate comparison a lower broken line curve has
been added for (b) calculated on the second basis (for (a) there would
be no difference).

Two important considerations, to some extent interconnected,
may be introduced here. The curves given relate, where possible, to
bodies of studiedly smooth surface in streams comparatively free
from turbulence. By turbulence is deliberately meant, as before, an

100 AERODYNAMICS [CH.

unsteadiness that is finely grained in comparison with the size of the
body, and not comparatively large-scale fluctuations of velocity such
as eddying might, produce. We could make a smooth and steady
tunnel stream turbulent by interposing upstream of the body a
mesh screen of cords. Alternatively we could give it large and fairly
rapid fluctuations of velocity by operating quickly an electrical
resistance in series with the armature or field of the driving motor.
But the second form of unsteadiness is not turbulence. Again,
the natural wind is subject to considerable variation in magni-
tude and direction of velocity, but except very close to the ground
it is free of turbulence in comparison with all ordinary tunnel
streams.

Now the initial turbulence in a stream approaching a body is found
to affect drag (or lift, etc.), particularly at scales where f(R) changes
sharply. The sphere at circa 3 X 1C6 affords a good example ;
change is here so sharp that drag actually decreases with increase of
speed, as occurs also with the circular cylinder at much the same
Reynolds number. In Fig. 40 (a) the full-line curve shows the

variation in CD( = D/^pF1^, for diameter d) against R ( = F<J/v)

for smooth flow, as obtained by towing a sphere freely through the
atmosphere. A notably turbulent tunnel would give, on the other
hand, the left-hand dotted curve ; the shaded area indicates common
variation in different tunnels. Similar remarks could be made in
connection with the example (c). We note immediately that at
critical scales (1) tunnels of different turbulence will disagree with one
another, (2) free air values oif(R) may differ from those determined
in any tunnel. Furthermore, if in a given tunnel size be greatly
varied, the relative scale of the turbulence will change and
the tunnel will disagree with itself, though R remain constant.
Thus in general a f(R) curve in a given tunnel is a narrow band
of readings ; the same curve for many tunnels would be a wider
band.

Effects on drag are expected to be produced chiefly by the tur-
bulence in parts of the stream that pass the body closely. To make
an initially smooth stream effectively turbulent, our mesh screen
might be reduced to diminutive proportions if suitably located. In
the limit it might be replaced by a thin wire bound round the fore-
part of the body. We infer (and may check by direct experiment)
that roughness of surface modifies /(/?). This is to be expected,
since geometrical shape is changed, but it may be noted that very
slight roughness may increase drag remarkably — e.g. with wings at

WIND-TUNNEL EXPERIMENT

101

in]

R > 106. Thus aerofoils are often polished or plated if they are to
be tested at large Reynolds numbers.

While postponing further investigation of the foregoing, we may
note that from the engineering point of view knowledge off(R) in a
given case may not be necessary. The engineer is commonly faced
with inadequate data which he must extrapolate, whatever the risk,
to a flight scale. A controlled turbulence in the tunnel may ease in
some cases this difficult process, though introducing artificiality at
model scale. By convention the degree of turbulence in a particular
tunnel is gauged by the Reynolds number at which CD for spheres is
0-3 (= 385,000 in the atmosphere). Spheres are supported from
the back to obviate effects
of attached wires ; the drag
can be inferred from the
pressure at the back.

Scale Effects (b)

The second meaning
' scale effect ' is of a more
applied nature and reserved
chiefly to wings and the like.
The aeroplane wing some-
times preserves a single
shape, but always assumes
various attitudes in course
of flight. Accordingly, a
wide view must be taken of
its performance before choice
can be made for a particular
craft. Features of engineer-
ing interest include maximum
lift coefficient, maximum
lift-drag ratio, the drag co-
efficient at certain small lift
coefficients, etc., and at what
incidences these occur is often
immaterial . Rayleigh's
formula can only, of course,
be applied to a given wing shape at constant incidence, so that in prac-
tice we have to deal with a series of f(R ) curves for a single wing. Now
it always happens that scale effect is more advantageous at some

0° 5°

INCIDENCE, a

41. — SCALE EFFECT ON L/D

GOTTINGEN 387 WING.
(a) 5-ft. low-speed wind tunnel corrected
to free air.

(6) Small full scale.

FIG.

10°

FOR

102

AERODYNAMICS

[CH. Ill

(a)

'RAF 48

incidences than at others, so that max. L/D, for example, occurs at one

incidence in the tunnel and at a different incidence in flight (see Fig.

41 , for example). The change from model to full scale of max. L/D,

max. CL, or other character-
istic, is then called scale
effect irrespective of modi-
fication of incidence.

Further examples * of
scale effect with this wider
meaning are given in Fig. 42.
The results shown at (a)
illustrate a difficulty that
will be now thoroughly ap-
preciated in interpreting
ordinary wind-tunnel results
in terms of full scale. As-
suming choice to be required
between the wings R.A.F.
38 and R.A.F. 48, and max.
lift to be the overruling con-
sideration, experiments at
R < 2 x 10e would give
preference to the second,
while actually it should be
given to the first.

Again, from the same
point of view, tests would
be required at R > 3$ X 10*
to decide definitely between
R.A.F. 38 and the wing
Gottingen 387. Neverthe-

FIG. 42. — APPROXIMATE SCALE EFFECTS

1-5

1-4

1-3

.20T

18C

14'

R.A.F 38

.GOTT387

J=48

7-0

Log,0R

MAXIMUM CL FOR THREE AEROFOILS (ASPECT
RATIO 6).

less, tests through a range of
much smaller Reynolds
numbers, within the com-
pass of a high-speed 6-ft. open-jet tunnel, would show max. CL
decreasing with increase of R for Gottingen 387. suggesting that
it might not maintain its great advantage at the small scale over
R.A.F. 38, whose max. CL would be shown increasing sharply. Such
evidence, though inconclusive, provides a better guide than com-
parison at a single small scale.

* These and other examples given are based on Relf, Jones, and Bell, A.R.C.R. &
M., 1627.

Chapter III A
EXPERIMENT AT HIGH SPEEDS

66. Variable-density Tunnel

High-speed phenomena in Aerodynamics are usually studied in
two parts : appreciably below the velocity of sound, or well above it.
Comparatively few quantitative measurements have been made for
Mach numbers within the range 0-9 to 1-3 owing to lack of sufficient
stability of flow. High-speed tunnels are consequently divided into
two distinct groups : the subsonic and the supersonic. Of these
the former is the more important, as exploring the compressible
flow that immediately precedes critical Mach numbers at which is
experienced shock, a phenomenon that is accompanied by great
increase of drag and decrease of lift.

One method of obtaining subsonic speeds for experiment is to
employ a tunnel of the ' race-course ' design partially evacuated.
Adaptation to withstand a crushing pressure of some 12 Ib. per sq.
in. permits alternative use under a bursting pressure of a few
atmospheres. The term variable density is often applied to tunnels
arranged only for compressed air, and justifiably so since their
pressure is varied, but it is gradually becoming reserved for those
in which the density can also be reduced.

In examining tunnels of this type we have to take into account
that at high speeds the density p, the absolute temperature T, and
the velocity of sound a in the experimental part of the stream are
much less than in the return flow just before contraction. The
latter state will be distinguished by suffix 1.

The power required to maintain a stream of section C and velocity
V, the power factor being defined as in Article 61, is

"•"•»• - 5155 '"••

If M denotes the Mach number V/at the formula may be rearranged

(i)

b.h.p. = M'.

F 1100

1100

103

104 AERODYNAMICS [CH.

by Article 30. Assuming a substantial contraction ratio, as is
specially desirable with this kind of tunnel, Vf can be neglected.in
comparison with V* and, introducing also the approximation
y = 1-4, (47) of Article 30 then gives

P f 1
-P = \ i _±

Pi I 6

i.e.,

^Pi
Substituting in (i),

M» .....

(111)

(1 + M'/5)* '

If we further assume P = J, TI = 288°, and write a for the ratio
of pt to the standard air density at sea-level, (iii) yields the approxi-
mate formula

b-h-p- -

Calculations by means of these formulae speedily show that a
large power is required to maintain a high Mach number in a stream
of moderate size in spite of considerable rarefaction. Thus for
M — 0-8, C = 100 sq. ft., and or = J, the h.p. required is nearly
8000 for the empty tunnel and would need further increase to cope
with the high drag of models under test. The large power, and the
consequently elaborate measures necessary to cool the relatively
small circulating mass of air, comprise the principal disadvantages
of this type of tunnel. On the other hand, the type possesses the
advantage of combining, in a single installation, provisions for high
Mach numbers at a reasonable experimental Reynolds number with
a complete model, and also for high experimental Reynolds numbers
when low Mach numbers can be accepted.

This can be verified as follows. Using (ii), the Reynolds number
R is given by

R ^

Now (ji is the coefficient of viscosity for air in the state prevailing at
the working section ; i.e. from (31) and (33) —

IJL = 3-72 X 10-' (T/288)8'4
where

HI A] EXPERIMENT AT HIGH SPEEDS 105

Hence, introducing <y as before and assuming ^ = 288°,

Referring to the numerical example above, R > 1 J million could
be expected with a complete model at M = 0-8. Again, putting
(j = 4 instead of J in (iv) leads for the same case to M = 0-28, and
the expectation of a Reynolds number of 11 or 12 million with a
complete model. The Mach number is then so low as to have no
significance, and the same Reynolds number could be realised with
25 atmospheres pressure in a tunnel of approximately half the size
and one-third the speed, entailing an expenditure of only some
450 h.p. But turning this small compressed-air tunnel into a
variable-density tunnel would enable only a small Reynolds number
to be reached with a complete model at a Mach number of 0-8, and
the h.p. would be increased to upwards of 2000.

It can now be seen that adjusting the density of the air used
for experiment presents opportunities for economy in two directions,
viz. by employing light air for high Mach numbers and heavy air
for high Reynolds numbers. Excessive use of either expedient is
to be avoided, however. A very low density leads to unacceptably
small Reynolds numbers at high Mach numbers, and a small mass
of attenuated air in which mechanical energy is being converted
into heat at a great rate is difficult to cool. Difficulties arising at
the other extreme have already b£en mentioned. Thus a com-
promise is sought between apparent economy, on the one hand, and
the advantages of moderate compressions and rarefactions on the
other. It is unfortunate, in view of the evident utility of the
variable-density tunnel, that these considerations point to a power
equipment of some 10,000 h.p. The number of such tunnels is
likely, therefore, always to be limited.

66A. Induced-flow Subsonic Tunnel

The imposing installations necessary to maintain experimental
streams of ordinary size at high speeds have led to the development
of small induction tunnels in which subsonic winds are produced
for short periods at a time by the exhaustion of reservoirs of com-
pressed air. Such tunnels are small, often about 1 sq. ft. in
cross-sectional area, and the limited supply of compressed air is
expended economically by use of the injector principle, a sheath of
high- velocity air from a pressure chamber entraining the flow of a
much greater volume from rest in the atmosphere. Reservoirs

A.D. - 4*

AERODYNAMICS [CH.

should be large and pumping plants powerful in order to provide
runs of sufficient duration at reasonable intervals, but only a
moderate pressure is called for.

These tunnels may have round, square, or other sections. A
vertical rectangular form has been developed at the National
Physical Laboratory for experiments on aerofoils under two-
dimensional conditions. Being of outstanding interest and suitable
for wide reproduction, it will be described * in some detail.

Fig. 42A indicates the main points of the apparatus. The section
is 17 J in. by 8 in., and the plane of the figure is parallel to
the wide sides. The downstream end of the tunnel is surrounded
by a distributing box or pressure chamber C, which receives com-
pressed air from a large reservoir and exhausts it through injector
slots J, about 4-2 millimetres wide, into a long divergent diffuser D.
The injector stream induces the flow of atmospheric air into the
flared intake A through a box baffle B of fine gauze screens which
raise the turbulence Reynolds number to the satisfactory value of
290,000 (cf. Article 65). The aerofoil stretches from one wide wall
to the other, and these walls are flat and parallel. The narrow walls,
opposite the upper and lower surfaces of the aerofoil, are made from
metal strip and can be adjusted separately to streamline forms, as
indicated by dotted lines in the figure, by means of closely spaced
micrometer screws. If they are shaped to lie along streamlines
appropriate to the aerofoil in the absence of walls, the condition of
free flight will be simulated, the permissible chord of the aerofoil
increased, and best use made of the restricted cross-sectional area
of the tunnel. Towards the outlet, the adjustable walls may con-
verge to a throat, where the stream attains to the velocity of sound
and creates a shock wave which has a beneficial effect in steadying
the flow upstream.

Mach numbers exceeding 0-9 can be produced by an injector
pressure of 80-90 Ib. per sq. in., which appears to be the optimum
pressure for this tunnel with the model in ; the blowing pressure
for the empty tunnel is 30-40 Ib. per sq. in. For measurements of
lift, drag, and pitching moment at the small incidences of interest in
connection with high Mach numbers, the permissible aerofoil chord
is 5 in. By (vi) of the preceding article with a = 1, this gives
a Reynolds number of 1-8 million at a Mach number of 0-8. An

* A preliminary description of this tunnel is included by kind permission of the
Aeronautical Research Council. The author's acknowledgments are also due to
the following, whose papers have been read:— Bailey and Wood, A.R.C.R. & M.
Nos. 1791 and 1853 ; Beavan and Hyde, A.R.C. Rept. No. 6622, 1942 ; and Lock
and Beavan, A.R.C. Rept. No. Ae. 2640, 1944.

EXPERIMENT AT HIGH SPEEDS

107

•g

H-j
Pi

III A]

aerofoil of more than
twice the above chord
can be tested at zero
incidence with only
manageable diffi-
culties, increasing the
Reynolds number at
need to 4 million.

Adjustment of
Walls. — For various
reasons these tunnels
are usually vertical,
but for brevity and
clarity the side op-
posite the upper sur-
face of the aerofoil
will be referred to as
the roof and that
opposite its lower
surface as the floor.
The roof and floor
alone are adjustable
in shape, the wider
sides remaining flat.
The approximately
free-air shape for a
non-lifting aerofoil is
determined in steps,
as follows.

The roof and floor
are first adjusted to
give constant pres-
sure (zero longitudi-
nal pressure gradient)
with the tunnel
.empty. Owing to
thickening boundary
layers along all four
sides and the com-
pressible nature of the flow, the required shape is neither simple
nor predictable, but it is easily arrived at by experiment.

Inserting the aerofoil varies the pressures along roof and floor

.
u 3

If
ftg

S %

fl -3

* 8

108

AERODYNAMICS

[CH.

from the constant values found for the empty tunnel, as illustrated
by curve (a) of Fig. 42B. The increment of speed at the edge of
the boundary layers of the roof and floor is much greater than would
exist at the same distance from the aerofoil in free flight and indi-
cates the difficulty experienced by the air in flowing past the
aerofoil in the presence of the tunnel. This is termed blockage ; the

peak effect opposite the
aerofoil is called solid
blockage, and the main-
tained effect behind, the
wake blockage. Clearly,
the back part of the
tunnel requires widening
in the presence of the
aerofoil. As a second
step, the roof and floor

FIG. 42B.-BLOCKAGE IN SUBSONIC TUNNEL. *** ^^^ adjusted in

(a) Pressure along ' flat ' wall ; (b) pressure along shaPe to g* ve constancy
streamline wall. of pressure along them,

the amount of readjust-
ment necessary being accurately noted from the graduated heads of
the micrometer screws used for the purpose. Finally, the roof and
floor are set to shapes approximately midway between those for
constant pressure with the tunnel empty and with it containing the
aerofoil, respectively (the factor O5 is for greater accuracy replaced
by 0-6).

The last step gives effect to a theoretical calculation by Taylor
and Goldstein concerning compressible flow between two parallel
walls, one flat and the other corrugated. This showed that, under
certain conditions, one-half of the pressure distribution along the
flat wall is caused by the corrugation of the other wall and one-half
by the constraint which the flat wall itself exerts on the flow.
Hence, a wall shaped to lie along free streamlines should exhibit,
under these conditions, one-half the pressure changes caused along
a flat wall by a disturbance. The roof and floor of the tunnel
adjusted to constant pressure with the tunnel empty are regarded
as flat in this sense, and linear variation of pressure change with
shape is assumed.

Blockage is then eliminated, the pressure distributions on roof
and floor being reduced to curve (b) of Fig. 42B, which accords
approximately with absence of constraint. A greater speed becomes
possible in the tunnel. But the essential point is that the flow past

Ill A] EXPERIMENT AT HIGH SPEEDS 109

the aerofoil should now be characteristic of free flight, a large
exaggeration due to tunnel interference having been removed.
A test of the sufficiency of the measures adopted is described in
Article 66C. Different settings of the roof and floor are required
for different aerofoils, or incidences, as well as for considerably
different Mach numbers with the same aerofoil.

The shape of the roof is the same as that of the floor for an
aerofoil of symmetrical section at zero incidence, to which the above
description applies, but is different if a lift exists. Free streamline
shapes for a lifting aerofoil involve bending the axis of the tunnel,
which is inconvenient, and an approximate process is adopted.
The principle of this further step will appear in later chapters, and
reference may be made to the papers cited for the best method of
application.

Methods of Measurement. — An Aerodynamic balance could be
used to weigh the force components on the aerofoil as with an
ordinary wind tunnel, except that it would require to be enclosed
or specially designed to take account of the large drop of pressure
in the tunnel. Forces are actually measured by wake exploration
for drag and by pressure plotting the aerofoil for lift and moment.
The lift of the aerofoil per unit length midway between the side
walls can also be determined by connecting a multiple-tube mano-
meter to a line of holes in the roof and in the floor and subtracting
the integrals of the pressure changes so recorded ; this known
method is illustrated quantitatively in Chapter VII, but obviously
the aerofoil lift must be supported on a long floor and roof by
pressure changes, and the only question arising is whether an
allowance must be made for restricted length.

66B. The Supersonic Tunnel

Raising the injector pressure of a subsonic tunnel of the type
just described fails to increase its speed appreciably. Tunnels that
provide wind speeds in excess of the velocity of sound are actuated
by connecting them directly to large reservoirs of compressed air,
or to receivers evacuated by pumps. Making extravagant use of
high-pressure storage space, they are kept very small, only a few
inches in width.

In Fig. 42c the reservoir or region of higher pressure is at O, and
a convergent nozzle N leads through a throat T to a long divergent
duct D, which is open to the atmosphere or region of lower pressure
at R. The location of quantities is distinguished below by use of
these letters as suffixes. Suffix 0 refers to air sufficiently removed

110 AERODYNAMICS [CH.

from the vicinity of the intake for its velocity V0 to be negligibly
small. When a supersonic tunnel is working satisfactorily the

entire flow, apart from a thin
boundary layer, is irrotational ;
the pressure is related to the
velocity by Bernoulli's equation
and to the density by the
adiabatic law. The boundary
layer will be ignored and the
tunnel assumed to run full with
constant velocity over each cross-
section.

For p0 little greater than pR
the apparatus works as a sub-
sonic tunnel or venturi tube, a

large pressure drop at the throat where the speed is a maximum
being mostly recovered in the divergent duct, along which the
kinetic energy is gradually reduced. This state is indicated
schematically by curve (a) in the figure. Increasing p0/pR at
first merely increases the pressure drop at the throat and the
speed. But a limiting condition is reached when VT == #T, the
velocity of sound in air of the low temperature then attained at
the throat. The corresponding minimum value of pT/p0 follows
immediately from (50) of Article 30 as —

FIG. 42c. — SUPERSONIC WIND TUNNEL.

mm. £- = mm. (
Po

y- (r

= 0-527.

(i)

Any further increase of p0/pR fails to produce a larger pressure drop
at the throat or to make VT exceed aT. But if p0— pR be made
substantially greater than 0-473^0, further expansion of the air
occurs along a suitable divergent duct in place of the former com-
pression, as indicated schematically by curve (6) in the figure.
Supersonic speeds result from the air expanding more rapidly than
the cross-sectional area of the duct. The expansion further reduces
the temperature of the air, and therefore the local velocity of sound,
by (27). Hence the significant ratio VD/aD is increased on two
counts. This ratio will be denoted by M.

Wide scope exists provided p0/pR can be made large. For
VQ = 0, (51) of Article 31 gives

III A]

EXPERIMENT AT HIGH SPEEDS

111

and, remembering that the second term in the curly brackets is
equal to (&vla0)*> the expression can be changed to

with the understanding that M applies to the position along the
duct at which the pressure is pD. Putting p0/pD = 10 and 20, for
example, yields M = 2-15 and 2-60, respectively. Greater values,
between 3 and 4, have been obtained in practice. At the other
extreme, M = ^/2 appears as a matter of recent experience to be
approximately the lowest Mach number at which the flow through
a supersonic tunnel is sufficiently stable for accurate experiments

to be made, and it gives p0lpD = yv ~l = 3J.

Let ST be the cross-sectional area of the throat. For any value
°* Pol PR greater than that required to secure FT = ar, the mass flow
per second through the tunnel remains constant. It will be denoted
by C, and from (i)

C =

(iii)

Now the mass flow through every cross-section must be the same.
Hence, neglecting the boundary layer,

Substituting for C from (iii).

This result applies to all sub-
sonic tunnels of the type con-
sidered and is plotted in Fig.
42D. An approximate value
for the constant coefficient is
14'7. To find the expansion of
the tunnel area required for a
given value of M , folpo is first
found from (ii) and then SD/5T
from(iv). The particular manner
in which it is chosen to expand
the tunnel beyond the throat will

FIG. 42D. — PRESSURE ALONG SUPERSONIC
TUNNEL.

112

AERODYNAMICS

[CH.

<-Throat

Distance downstream

FIG. 42B. — SHOCK WAVE IN SUPERSONIC
TUNNEL.

then fix the position of the experimental station along the divergent
duct. Correction is required on account of the boundary layer.

Downstream of the experimental station the duct expands further
in some continuous manner to a maximum cross-sectional area S'
before discharging into the atmosphere or low-pressure receiver.
The flow can remain irrotational throughout the length only if the
pressure p' at S' also satisfies (iv). A shock wave is likely to occur
somewhere within D if pR > pf , and travel upstream towards the
throat if pR is increased further. In Fig. 42E, (a) is the continuous

pressure curve along a tunnel
in which />R is not greater than
p' ; (b) is the discontinuous
pressure curve appropriate to a
shock wave forming at B ; (c)
shows an earlier failure due to a
still greater value of />R. Irrota-
tional flow is possible only as far
as the wave front. If pR < p' ,
M will not be increased, but a
shock wave can be expected in
the issuing jet.

It will be seen that only one value of M is possible at a fixed
experimental station along a given tunnel. Displacing the station
along the divergent duct is usually inconvenient. But the small
size of the tunnel enables large variations of M to be obtained
readily by employing alternative divergent ducts, provided po/fa
is sufficiently large. From this point of view, and also to lengthen
the necessarily brief time of a run even at a comparatively small
Mach number, the reservoirs (in the common case of tunnels exhaust-
ing into the atmosphere) should be capable of withstanding air
pressures of many atmospheres and possess a large capacity,
suggesting a battery of long boiler shells of moderate diameter
together with pumps of substantial power. But for visual purposes
in a small laboratory a modest equipment will provide a very small
supersonic stream for, say, a minute at a time. For quantitative
two-dimensional work with larger apparatus, the section will be put
to best use if made deeply rectangular in shape, or fitted with adjust-
able sides as described for the N.P.L. subsonic tunnel.

66C. Illustrative Results

Published experiments of aeronautical interest at high Mach
numbers are sparse and largely confined to two-dimensional tests.

Ill A] EXPERIMENT AT HIGH SPEEDS 113

The illustrations now given are no more than typical in a qualitative
way.

As a first example, the left-hand side of Fig. 42F gives the drag
curves for certain symmetrical aerofoils of three different thickness
ratios (maximum thickness of section expressed in terms of the
chord) tested at zero inci-
dence through a range of
high subsonic speeds. CD
remains almost constant c
until, at some critical Mach
number depending on the
section, there occurs what uos o* 07 os 0-9 ro — ^ rs KM? re w 2

is known as the compressi- rt „

, ., ., , . ^ „ * FIG. 42F. — CRITICAL MACH NUMBERS.

Mtty or shock stall. There- The curyes (a) (fe) (c) are for 8ymmetrical

after CD increases very aerofoils in ascending order of thickness.

rapidly ; some tests suggest

to five or more times its value for incompressible flow. Supersonic
tests are difficult to carry out for M < 1-4, as already mentioned,
but it is clear that after a peak value CD decreases as indicated
qualitatively on the right-hand side of the figure, though not to its
small initial value.

The determination of the critical Mach number is evidently a
matter of first importance and likely to be affected by the presence

of tunnel walls. Fig. 42c
reproduces some of the results
of an investigation * of the
reliability in this connection
of the small N.P.L. tunnel
described in Article 66 A.
Curve (a) was obtained with
an aerofoil chord of 12 in.
and curve (c) with one of
5 in., both with streamlined
walls. Curve (b) applies to
12-in. chord with the walls shaped to give constant pressure in the
absence of the aerofoil. Comparison shows the magnitude of the cor-
rection achieved in a rather extreme case (the depth of the tunnel
being only 17 J in.) by streamlining the walls, and the fair agreement
reached as to the critical Mach number for the given aerofoil section
with a test employing a much-reduced chord, for which the corre-
sponding correction is relatively small. Thus the test on the
* Lock and Beavan, loe. cit.> p. 106.

0*65

0-85

FIG. 42c. — CHECK ON STREAMLINING
WALLS.

114

AERODYNAMICS

[CH.

>-,

INCOMPRESSIBLE FLOW

5 Oti 07 0-8 0'9 10

TV! 15 16 17 18 19 2

FIG. 42n. — SHOCK STALL.

smaller aerofoil can be regarded with some confidence as approxi-
mating to free-air conditions. Apart from its immediate interest,
the investigation illustrates the care that should be taken to
establish the validity of wind-tunnel experiments in general.

Fig. 42H illustrates the nature of the lift curve obtained for a
fairly thin cambered aerofoil at a small angle of incidence. CL
increases strongly at first, but the shock stall causes a rapid loss of
lift. The initial increase is of special importance in the design of

airscrews. Beyond the
velocity of sound, recovery
of lift is poor, drag remains
high, and the maximum
lift-drag ratio obtainable
with any aerofoil has a
much-reduced value.

It is possible to gain a
preliminary idea of the
nature of the shock stall

by visually inspecting the flow under suitable illumination ;
for this reason high-speed tunnels may be fitted with glass sides
opposite the aerofoil. The critical Mach number is associated with
the generation of a shock wave, which extends from the aerofoil
for some distance into the stream and casts a shadow.* The wave
changes the Bernoulli constant for flow passing through it, creating
the compressibility wake already noted. With further increase of
Mach number the wave penetrates more deeply into the stream and
becomes rapidly displaced backward. In a tunnel, a stage is soon
reached when it extends right across to the roof, and experiment
becomes more difficult. The stall is easily caused to occur early
by employing a bluff section ; e.g. a circular cylinder gives a shock
wave at circa M = 0-45. It can also be delayed to some extent
by reducing thickness ratio, as illustrated in Fig. 42F, and by
shaping the profile to minimise the maximum velocity attained by
the wind in flowing past. At supersonic speeds a shock wave is
formed in front of the body. These considerations apply not only
to models but equally to any exposed attachment used to support
them in the stream or to explore the flow in their vicinity.

66D. The Pitot Tube at Supersonic Speeds

The speed of supersonic tunnels is inferred from the static pressure
drop at the experimental station as obtained from a hole in the wall.

* See frontispiece.

Ill A] EXPERIMENT AT HIGH SPEEDS 115

However, a pilot tube may be used in experiment, and then a
special formula is required to deduce the speed from its pressure P,
the shock wave formed in front causing P to differ considerably
from the pitot pressure for irrotational compressible flow. If p
denotes the static pressure of the oncoming wind upstream of the
wave, the formula is, in approximate terms,

/p\2/7 / \2/7

(-) = 0.6,7 (MI.-I)

To much the same approximation, (52) of Article 31 gives for irrota-
tional flow

\pl 5

and the loss of pitot head implied in (81) becomes large at Mach
numbers considerably greater than unity. The formula may also
be used to estimate the speed of an aircraft diving at a supersonic
speed, and the pressure at a front stagnation point.

The theory is due to Rankine and Rayleigh and is summarised
below, partly in view of the importance of the case and partly as
illustrating, with a minimum of analytical complication, the nature
of the phenomena occurring at very high speeds. Simplification
arises from the legitimate assumption that though the wave is in
fact flatly conical, only the bluntly rounded apex of the cone,
immediately in front of the mouth of tfre tube, is likely to be effective,
and that this may be treated as plane and normal to the direction
of motion.

Rankine's Relationship. — Fig. 42i shows, in front of the pitot
tube, part of an infinite plane shock wave which is stationary and
normal to the wind. Through the very small
thickness the velocity of the air is diminished
from V to Vl9 and its pressure and density
increased from p, p to pv pr Up to the A A c
wave and beyond it the pressure and density J

are related by the adiabatic law, i.e. the
flow is isentropic, but the air increases FIG. 42i.

in entropy on penetrating the wave.

Consider unit area of the wave, and let m be the mass of air cross-
ing this area per second. In doing work against the increase of
pressure at the rate

pv-

kinetic energy is lost at the rate

116 AERODYNAMICS [CH.

and the air also loses internal energy at the rate

*• fP Pi\

r - 1 \ p P!/ '

by Article 30. The principle of the conservation of energy demands,
therefore, that

pV - PlVl + lm(V* - Fx') + -^ (t - ^) = O (i)

Now the increase of pressure is equal to the rate of loss of momen-
tum per unit area, i.e.

Pl - p = m(V - VJ, . . (ii)
so that the loss of kinetic energy can be expressed as

i(A - P) (V + v,) = ~(pv- pw + \(p + A) (V - FO,

and since

m = pF = PlFx, . . . (iii)

Hence (i) reduces to

This is the Rankine,* or Rankine-Hugoniot, relationship between
the pressures and densities on the two sides of the wave. It readily
gives

PI _ (Y - i)# + (Y + i) Pi (v)

P (Y + \)p + (Y -

or approximately —

Pi WilP) + 1 Pi 6(pi/p) ~

P (PilP) + G' P
The Pitot Pressure. — From (ii) and (iii),

whence

F* oF8
M« » _ = il-

y/> ' 1 _

* Phil. Trans. Roy. Soc., vol. 160. 1870.

Ill A] EXPERIMENT AT HIGH SPEEDS 117

On substituting for p/px from (v), this gives

• (vi)
In like manner

M , = *V _ Pi*V

1KX -I — — —

+ ^+T-1]- • <vii>

Now, the pitot pressure P can be obtained by Articles 30 and 31 in
terms of quantities on the far side of the wave as

since the flow between the wave and the pitot tube is isentropic.
Equating (vii) and (viii),

/ P\ y v + 1 T v p

(p) -Vlfr-I)A + T +

Y-l

and, multiplying through by ( ^J , we have finally

p 4Y p

M is obtained from pjp by (vi). With the approximation 1-4 for y,

p 6

and (ix) reduces to (81), closely.

* Rayleigh, Proc. Roy. Soc., A, vol. 84, 1910.

Chapter IV
AIRCRAFT IN STEADY FLIGHT

67. Aircraft

Examples of airships have been given in Fig. 7. Heavier-than-air
craft are illustrated in Fig. 43. These depend for lift entirely upon
motion through the atmosphere either as a whole, as with aeroplanes,
seaplanes, and flying boats, or in respect of their lifting surfaces
which then have a relative motion, as with autogyros and helicopters.
Description of the latter type must refer to Airscrew Theory, and so
is deferred. Investigations of the present chapter are for the most
part expressed in terms of the aeroplane, but apply equally to the
seaplane and flying boat with modifications in detail only. Many of
the principles established also apply in a general way to autogyros
and helicopters.

All heavier-than-air flying depends first and foremost on the lift
of wings of bird-like section, which has already received preliminary
discussion (cf. Article 46) and the theory of which, the subject of later
chapters, mathematically resembles that of the electric motor. The
utility of wings of this kind was first realised by Horatio Phillips and
their principle by Lanchester. Most aircraft avoid flapping as a
matter of structural and mechanical expediency. Aircraft wings
are much more heavily loaded than those of birds to improve per-
formance and minimise structural weight. Useful flying depends
acutely upon extremely light power plant. Specialised development
of reliable engines has been remarkable, and large units, complete
with metal airscrews of variable pitch, now weigh less than l\ Ib.
per brake h.p. It may be said that, following the work of the
pioneers, high-performance flying first waited upon engine design,
then upon Aerodynamics,* whilst now further improvement is
equally concerned with Aerodynamics and new methods of
propulsion.

No attempt is made in this book to describe the work of the pioneers
of aviation ; even a cursory record of their gallant and brilliant
achievements would occupy much space. But in no department

* Cf. Relf, Institution of Civil Engineers, James Forrest Lecture, 1936.

118

CH. IV] AIRCRAFT IN STEADY FLIGHT 119

were their experiences more valuable than in relation to means of
control, and here especially, perhaps, may tribute be paid to the
Wright Brothers. Safety in the air is a first consideration and
depends notably on an inherent stability of the craft, tending to
conserve any of the various forms of flight to which it may be set,
and also on the provision of adequate means of control.

There has long existed a successful Theory of Stability due to
Bryan and Bairstow. The layouts of the craft illustrated, in regard
to the proportions and positions of the stabilising surfaces considered
in conjunction with the location of the centre of gravity and the
moments of inertia about the three principal axes, express a con-
venient and usual (though not the only) manner in which the
principles established are given effect.

Control surfaces, adjustable from the pilot's cockpit, are indicated
and named in Fig. 43. The rudder, attached to a fixed fin, gives
directional control in a familiar way. Orientation of the craft in
plan to the flight path is known as yaw, and angular velocity, pro-
ducing a change of yaw, as yawing. Horizontal rudders, called
elevators, are hinged to a fixed or only slowly adjustable tail plane,
and control attitude or incidence to the flight path in side elevation.
This is termed pitch, and angular motion that varies pitch is called
pitching: The ailerons move differentially, rolling the craft about
its longitudinal axis. A fourth control is provided by the engine
throttle.

68. An aircraft cannot maintain exactly a steady state of motion.
Disturbances arise from many causes and a continuous adjustment
takes place through either its inherent stability or judicious use of
controls by the pilot. Flight consequently proceeds in a series of
oscillations or wide corrected curves. Nevertheless, whether by
stability or control, uniform flight is closely approximated to for
short periods under favourable atmospheric conditions. It is
assumed to persist in the present chapter.

Study of an aircraft in these circumstances has for immediate
objects the determination of equilibrium and control and the estima-
tion of performance. These enquiries can quickly assume a rather
complicated character, and only first principles will occupy us for
the present. Early study of the elements of flight is desirable,
however, to obtain a general view which will be a guide to the
practical aims of the theories that follow in subsequent chapters.

Various assumptions are introduced in order to avoid detail.
It is assumed, for instance, that the aircraft in straight flight has a
plane of symmetry, a characteristic that can hold exactly only in the

120

•~ a

121

122 AERODYNAMICS [CH.

absence of airscrew torque. Flight in this plane is known as sym-
metric flight ; roll, yaw, rolling, yawing, and crosswind force must
all be absent. Asymmetric flight, which includes such common
motions as turning and side slipping, can also be uniform.

Some simple unsteady motions are referred to briefly, but adequate
study of manoeuvring, and the transient air loads to which it gives
rise, is postponed.

69. Except for temporary purposes, Aerodynamic lift is un-
necessary for airships, and the investigation of their equilibrium
and performance is consequently straightforward. The continuous
generation of Aerodynamic lift by aeroplanes and flying-boats, on
the other hand, results in peculiarities which have no counterpart in
other forms of transport. These characteristics are implied in the
standard coefficients determined from experiments on models in
wind tunnels, which readily suffice to reveal the main features of
aeroplane flight, and are used both in the present chapter and in
technical performance calculations. But a preliminary discussion
in more general terms introduces an alternative method, which,
though of less technical accuracy, has the advantage of explaining
the reason for the above distinguishing features.

The duty of an aircraft is to carry a large useful or disposable load
from one place to another quickly and at low cost in fuel. The
tare weight Wt and the drag D should clearly be minimised in
comparison with the lift L, provided the true air speed V is not
unduly decreased. High speed is especially necessary for aircraft,
since the velocity of every head wind must be subtracted in full ;
and it is also their prerogative, being most economically and safely
attained in their case.

Consider two series of geometrically similar aircraft, a sequence
of airships and another of aeroplanes, in straight and level flight, so
that in every case L = W, the total weight. Denote size by /,
and let this be sufficiently large for the materials of construction to
be used economically. Then approximately, Wt oc /3, though in
practice this relationship is considerably affected by variations in
requisite structural strength and by ' fixed ' weights, i.e. those of
components or equipment which depend little on aircraft size.

For the airships, assuming the same gas and a constant ceiling,
L oc /*, whence Wt/L is approximately constant.

The lift of the aeroplanes depends on speed as well as size, but
is equal to Sw, where S is the wing area and w the wing-loading
W/S = LjSt in straight, level flight. Since L/w oc l\ Wt/L will be
constant in their case only if w oc L1/3. This slow increase of wing-

IV] AIRCRAFT IN STEADY FLIGHT 123

loading with lift entails faster landing speeds for big aircraft, as
will be investigated later, but is evidently not an unreasonable
requirement within limits. It may be mentioned at once that in
1903 the loading per square foot of wing area was 2 Ib. (the Wright
biplane), by 1933-4 (the end of the biplane period) it had reached
15 Ib., and a year or two later J cwt., whilst now wing-loadings of
about \ cwt. are in use and f cwt. per square foot are contemplated.
As a matter of experience, aeroplanes or flying-boats exceeding
50 tons in weight can realise as small values of WJL as can airships
of 2-3 times the weight. Aeroplanes cannot indefinitely increase
in size as, theoretically, can airships, but the disparity in gross
weight between practicable airships and the largest aeroplanes
capable of realising acceptable values of Wt/L is decreasing. Thus
it is reasonable to compare the two types on the basis of lift.

Neglecting Aerodynamical scale effects, the drag of the airships
is given by —

D = CpFa/a, (i)

where C is a non-dimensional coefficient and constant for the shape
concerned. D\L oc pF2/7. But pF2 oc p0F,a, Po being the standard
density of the air at sea-level and V, the indicated air speed. Thus
alternatively D/L oc F,2//. The evident advantage of increasing
size arises geometrically from the linear reduction of the ratio of
surface area to volume. The fact tha^t small indicated air speeds
give very small values of D/L is without interest because of head
winds. The question of interest is : At what speed (if any) does
D/L become prohibitively large ?

Turning to aeroplanes for an answer, we have first to note that
only part of their drag, called the total parasitic drag DP, can be
expressed in the form (i). This form is also restricted, as will be
illustrated in due course, to the upper two-thirds of their speed
range owing to increased form drag at the large incidences necessary
for lower speeds. The remaining part of the drag, viz. the induced
drag Df (Article 59A), arises in a complicated manner and takes an
entirely different form. Adequate investigation must be post-
poned, but the principle underlying its peculiar nature may readily
be seen by reference to an artificial system in which the action of
the wings in generating lift is represented as imparting a uniform
downward velocity v to all elements of a mass m of air per second,
so that L = mv. In the actual system, lift is derived by the same
principle, but the air flown through is affected unequally.

This action communicates kinetic energy to the atmosphere at

124 AERODYNAMICS [CH.

the rate \mv*t which must be equal to the rate of doing work against
Dit i.e. to D4V, whence —

D( = ^v». . . . (ii)

Ignoring the effects of viscosity, the velocity v is essentially residual
and cannot come into being suddenly at the wings ; we must
assume that a pressure field, travelling with the wings, starts the
mass into motion some distance in front and leaves it with the
velocity v only at some distance behind. Let v' be the uniform
downward velocity of the mass m in the vicinity of the wings, and
assume v'jV to be small. Then another expression for the induced
drag can be constructed from the reflection that it must be equal
to the resolved part of the Aerodynamic force on the wings, which is
sensibly equal to the lift. The alternative expression is —

_ v' v'

' = v = vmv'

Comparison with (ii) gives v' = \v, and the second expression
becomes —

A = i~L' ' ' ' (iii)

Considering change of size and speed with constant shape, the
volume of air affected each second, viz. w/p, oc F7f, whence (ii) can
be written D4 = \k$v*l*t and combining with (iii) gives —

v ~

Substituting in (iii) and writing A for 1/2&,

When it becomes possible to take the unequal motion of the air
into account, (iv) will be verified to have the correct form. Thus
the formula for the total drag of each of these similar aeroplanes
may be taken as —

D = D< + DP = - + BpV#, . . (v)

where A and B are constant coefficients in so far as Aerodynamical
scale effects and incidence effects on form drag can be neglected.
For any aeroplane of the series, / and L are constant in straight,

IV] AIRCRAFT IN STEADY FLIGHT 126

level flight, and differentiation with respect to pF1 gives the drag
to be a minimum for that aeroplane when —

-. —( — }

and — minimum D = 2L\^ABt . . . (vii)
so that — minimum D/L = 2^/AB. .... (viii)

It is useful to notice that, since pFa = p0F^(22/15)1, (vi) gives
very closely —

1/4 //A 1/3

/ .4x1

* = 14 (B)

This indicated air speed for minimum drag will be denoted by F|0.

Thus the essential peculiarity of an aeroplane or flying-boat as
a means of transport is that minimum drag occurs at a certain
intermediate speed ; in other words, that D actually decreases
when V increases, so long as Vi < Ft0. For a given shape of
aeroplane, /a is proportional to the wing area, and (ix) can be written
in the form —

F.-o-to1/2. (x)

Thus we have that the minimum value of Z)/L is a constant for that
shape, and the speed at which it is realised can be adjusted in so
far as w can be varied. Subsequent calculations will show that
limitations imposed on the increase of w keep Fi0 too small for low-
altitude flying, judged by modern standards of aircraft speed,
though it is much larger than the speed of a 200-ton airship (about
100 m.p.h.), for which the value of D\L would be the same. Aero-
planes fly faster than F,0, incurring considerably more than the
minimum drag, but putting to use the whole, or part, of a large
margin of engine power which must in any case be carried in order
to provide for doing work against gravity at a sufficient rate to gain
altitude quickly when required.

When Vi is much greater than Fio, the first term on the right of
(v) becomes small in comparison with the second, and the drag of
aeroplanes tends to become more nearly expressible in the form (i).
Assuming clean cantilever wings and rigid airship hulls, no great
difference exists between the coefficients B and C specified on the
surface area, but the aeroplanes have less values of DjL than would
airships at such speeds because their surface area for a given lift is
much smaller. A further advantage to be derived from increasing
wing-loading is thus perceived.

126

AERODYNAMICS

[CH.

The first term on the right of (v) predominates, on the other hand,
when Vi is much less than Ft0 ; if aeroplanes could be designed to
fly really slowly, their drag would become prohibitively large.
Aeroplanes become inferior to airships on the present basis at
speeds less than 100 m.p.h. (or 75 m.p.h., if small airships are
admitted for comparison). The reason is partly that already
stated and partly due to B becoming greater than C when, in order
to economise in the weight of large lightly loaded wings, the clean
cantilever design suitable for substantial wing-loadings gives place
first to external bracing and finally to biplane design.

70. Airship in Straight Horizontal Flight on Even Keel

A rigid airship can be trimmed by movement of ballast or fuel, a
dirigible balloon by transference of air between forward and aft
ballonets. On an even keel there is least resistance to motion. Let
this drag at velocity V relative to the wind be D, W the total weight,
Lf the gas lift, T the resultant thrust of the airscrews. For steady
rectilinear horizontal flight —

W =L't T =D
and T satisfies —

TV
550

= H

where H is the thrust h.p., i.e. the total b.h.p. of the engines x the
efficiency of the airscrews. It is also required that no resultant
couple act. The centre of buoyancy B, (Fig. 44), is above the centre

FIG. 44.

of volume of the envelope ; the C.G. (G) is low, but possibly above
the line of action of T ; D is the sum of the drags of envelope, tail
unit, gondolas, and airscrew struts, and its line of action is appreci-
ably below the centre of volume, because the envelope and fin drag,

IV]

AIRCRAFT IN STEADY FLIGHT

127

acting axially, constitutes only 80 per cent, of the whole. Taking
moments about G and using the notation of the figure —

Dd — L'x = 0
D(t + d) = Wx.

If there is no tail lift, G is forward of B, but only slightly in a practical
case. For example, W might be 150 tons, t + d 40 ft., T = D =
15,000 lb.; when x = 1-79 ft., or 0-25 per cent., perhaps, of the total
length.

The drag coefficient varies in a complicated manner through the
very wide range of Reynolds number (7?) occurring in practice (from
0 at zero speed to 6 x 108, if length of hull be used in specifying R).
Direct model experiment can give only a rough estimate of full-scale
drag ; this is matter for semi-empirical theory and full-scale experi-
ment. From 15 to 25 b.h.p. per ton are usually supplied.

71. Airship Pitched

Now consider steady straight horizontal flight, but with the air-
ship pitched nose up. Fig. 45 gives the normal pressure difference

0-J

pv2

(BOTTOM)

10°(TOP)

-0-1

NOSE

TAIL

FIG. 46. — PRESSURE DISTRIBUTION ALONG AIRSHIP.

along the top and bottom of the hull of Fig. 7 (c), when level and when
pitched at a = 10°, showing Aerodynamic lift (L) in the latter case.
Associated with this is an Aerodynamic pitching moment M . Re-
ferring to Fig. 46, x has increased owing to the pitch. An Aero-

128

AERODYNAMICS

[CH.

FIG. 46.

dynamic force Lit exerted by the horizontal fins and elevators, acts
at a distance / behind G, maintaining the pitch. From Fig. 46 :

W = L' + L + Lt + T sin a
T cos a ~ D

Tt + Dd + M -L'x- Ltl = 0.

The lengths, etc., denoted by these symbols are not the same, of
course, as in the preceding article, but T must satisfy the same h.p.
equation as before.

With increase of a, D tends to increase, so that V must diminish.
Thus L increases on account of a, but decreases on account of 7, and
a maximum value will evidently occur at some particular a and
corresponding V, assuming the elevators to be sufficiently large to
permit the last equation to be satisfied.

The airship illustrated in Fig. 1 (c) had L' = 157 tons and maxi-
mum b.h.p. = 4200 ; the curve of possible Aerodynamic lift against

speed has been estimated as in Fig.
47. It appears that L may here
exceed 12 per cent, of £', but the
large decrease in speed will be
noted. With less engine power
available this maximum percentage
would be less.

Little speed is lost, on the other
hand, at a small angle of pitch,
giving, for example, one-third of
the maximum Aerodynamic lift.
Airships fly cabrS (tail down)

o 10 20 so 4O commonly for three reasons : (1)
SPEED Loss (m.p.h.) decreased gas lift, resulting from

Fia. 47. either general loss of gas or consider-

O-15

IV] AIRCRAFT IN STEADY FLIGHT 129

able and sharp change of temperature, (2) transient overload at the
beginning of a long flight due to fuel, (3) failure of a gas-bag. In the
last case the shift of the centre of buoyancy may be sufficient,
depending upon the fore and aft position of the fault, to prevent the
elevators from holding the craft to the required pitch for equilibrium.

72. Aerodynamic Climb of an Airship

While the gas-bags remain only partly full, an airship can be
steered to higher altitudes. If ballast is discharged during flight at
zero pitch, the craft rises until the gas-bags fill, and an equal mass
of gas is valved. Temperature lag in the gas, described in Article 17,
results in slow attainment of ultimate altitude. Thus, rapid climb
to a given altitude by discharge of ballast entails subsequent slow
ascent, with a loss of gas that may be needless. Such waste is
minimised by Aerodynamic climb, when positive pitch to the up-
wardly inclined flight path provides Aerodynamic lift, supporting
excess ballast until the gas has time to complete expansion appro-
priate to the new pressure and temperature.

The conditions for steady climb at any instant are simply stated.
Resolving along and perpendicular to the path, inclined at 6 to the
horizon —

L' cos 6 + L = W cos 6 >

L' sin 6 + T = Wsin Q + D,

since L is perpendicular to the direction of motion. During such a
climb L must gradually be increased, however, to compensate for
decreasing gas lift. The engines now do work against gravity in
respect of the excess ballast.

73. Aeroplane in Straight Level Flight > ( l * h ' ^'l'1^

The vertical position of the C.G. of a heavier-than-air craft varies
considerably with type, but longitudinal position is restricted by

FIG.

A..D. — 5

ISO AERODYNAMICS [CH.

Aerodynamic conditions. In the normal case, travel of the centre
of pressure, already described, leads to unstable moments about the
C.G., which require to be counteracted by the tail plane.

Fig. 48 refers to a low- wing monoplane of weight W acting at G.
A is the resultant Aerodynamic force on the whole craft, excluding
the airscrew thrust T and the tail lift Lt. Thus, with these ex-
clusions, L is the lift and D = L tan y is the drag of the whole craft.
It is assumed that crosswind force and couples about vertical and
longitudinal axes vanish. Then for steady horizontal rectilinear
flight at velocity V, with leverages as indicated in the figure —

W = L + Lt + T sin p . . . (82)

T cos p = D ..... (83)

Aa=LJ + Tt . . . . (84)

with r=550#/F . . . . (85)

where H is the total thrust h.p. as before.

First Approximation. — The above equations present no diffi-
culties given adequate data, but they are complicated by technical
detail. A first approximation follows the assumptions : (1) that a
is small compared with / and that Lt can be neglected in comparison
with L, (2) that the sum total of the lifts of all components of the
craft other than the wings and tail plane can be neglected in com-
parison with the lift Lw of the wings, (3) that p and t may be ignored.
Then the equations become :

W=LW=CL.^V*S .... (86)

T = D = 550 H)V .... (87)

Aa = Ltl ...... (88)

where S is the area of the wings and CL their lift coefficient.

In order to describe the primary characteristics of aeroplane
flight, we adopt these simplified expressions together with the further
approximations :

W = const. ...... (89)

where Dw is the drag of the wings according to data appropriate to a
single Aerodynamic scale within the speed range of the craft, r is
the lift-drag ratio of the wings only, DB is the 'extra-to-wing' drag,
i.e. the drag apart from the wings, and DB its value at standard
density and a particular speed V'f preferably within the range. It
will be observed that we neglect scale effects through the flying range
of scale. This applies also to the lift curve of the wings, but the

IV] AIRCRAFT IN STEADY FLIGHT 131

scale chosen in their case is at least that for the minimum flying
speed. In (89) we ignore loss of weight through consumption of fuel.
In (90) we also omit to take into account variation in airscrew slip-
stream effects ; these will be allowed for in estimating H.

If at constant altitude V changes from Vl to Vtt the corresponding
lift coefficients are related by :

CLa

provided S remains constant, which with present aircraft is im-
plied conventionally in CL. DB2 follows from Z)B1 by the relation

- • • • (92)

These expressions are independent of the shape of the wings or con-
stancy of that shape. But resulting values of wing drag and inci-
dence depend upon shape. If this is constant, r is conveniently
read from an r — CL curve ; if it is continuously variable for changing
flight, r may be read from the evolute of a family of such curves, one
for each shape, but the result will express an ideal that the pilot may
not quite realise in practice. Incidence is similarly determined.

Before the performance of any given aeroplane is examined, it is
necessary to know 5. Considerations affecting choice of area are
discussed in the following three articles.

74. Minimum Flying Speed and Size of Wings of Fixed Shape

While, as is either true or implied in CL, S remains constant,
from (86) V is a minimum for a particular craft in steady level flight
when CLp is a maximum, i.e. at low altitude when CL is a maximum.
The speed at which CL reaches its maximum value is called the stall-
ing speed ; further loss of speed leads to Lw < W and descent occurs.

Typical examples are given in Fig. 42 of full-scale maximum values
of CL for wings of fixed shape, according to the compressed-air tunnel.
Without special devices the value 1-5 for CL is not easily exceeded,
even in the case of large monoplanes for which R = 107 at minimum
speed. In terms of w = W/S, the wing-loading already introduced,
(86) becomes

z0 = CL.JPF* .... (93)

The following table gives, for various speeds chosen as minima,
approximate corresponding values of w, S, and span (on the basis of
an aspect ratio of 7), for W = 10 tons, assuming maximum CL= 1-60.

132

AERODYNAMICS

[CH.

V (ft. per sec.)

w (lb. per sq. ft.)

S (sq. ft.)

Span (ft.)

106

20-0

1120

11-4

1060

117

6-4

3485

156

The smallest span given may be regarded as roughly the greatest for
which a reasonably light wing structure of sufficient strength could
be expected without external bracing. To economise on wing
weight and for other reasons it is advisable, indeed, to have w > 20,
and often w exceeds 40 lb. per sq. ft. On the other hand, high
minimum speeds lead to danger in forced landings on unprepared
ground. Such comparisons lead to two general conclusions : Really
low stalling speeds cannot be designed for economically in aero-
planes, seaplanes, and flying-boats. Special devices to reduce such
speeds by adapting wing shape are important.

75. Landing Conditions

Reference to Fig. 42 shows that maximum CL may require the
incidence a of the wing to exceed 18°. Now a = 0, approximately,
for high speeds, when the fuselage or body should be horizontal iii
level flight, for low drag. The C.P. of the tail plane is usually distant
0-4 to 0-5 of the span behind the C.G. of the craft. Further, unless
a nose-wheel exists undercarriage wheels must be located consider-
ably in front of the C.G. to prevent overturning on the ground, due
to running the engines at full power with wheels chocked, or applying
brakes. Thus, to land at 18° would mean a high and heavy under-
carriage : 13° is often the economic limit.

Fig. 49 gives CL — a curves for the wing Clark YH illustrated in
Fig. 34 (aspect ratio 6) for a small aeroplane (5-ft. chord) with low
stalling speed (48 m.p.h.) — lower curve ; and for a larger craft of
higher stalling speed— upper curve. At the greater Aerodynamic
scale, CL drops from 1-48 at 18-3° to < 1-20 at 13°. The lift coeffi-
cient available for landing is apparently, therefore, considerably
less than the maximum.

This disadvantage may be offset by an increase that occurs in
maximum CL when a wing is in motion only a few feet above the
ground. (It will be recalled that an aerofoil usually gives an ap-
preciably greater maximum lift coefficient between the walls of an
enclosed-section wind tunnel than in an open jet.) The tail plane may
also contribute to lift.

A further correction exists in the hands of experienced pilots who

AIRCRAFT IN STEADY FLIGHT

133

1-0

0-8

0-4

10°

15"

20°

25°

30°

IV]

land aeroplanes with
great skill in an un-
steady motion, re-
alising landing
speeds which are
often lower than de-
signers have reason
to expect and which
scarcely exceed the
stalling speeds.

One other point
must be mentioned.
At stalling speed the
various Aerody-
namic controls of a
craft tend to become
inefficient. A pilot
therefore ' brings an
aeroplane in,' i.e.
descends prepara-
tory to landing, con-
siderably faster ; 20
per cent, excess over stalling speed is not uncommon, when by
(91) CL would have 0-7 of its maximum value, corresponding to
less than 11° incidence with the Clark YH wing, or 2° less,
perhaps, than the standing angle. Speed is still high after
flattening out the flight path to within a few feet above the
ground. Moreover, so far removed from the stall, the lift-drag
ratio is high. Little drag exists and the aeroplane tends to ' float/
i.e. to proceed a considerable distance before actually landing. Yet
it is essential with high landing speeds to make contact with the
ground quickly after flattening out, so that the brakes can bring the
craft to a standstill within the distance prescribed by the aerodrome.
It is desirable to have large drag on landing, and this, together
with reduction in speed of approach and a further advantage to
be described later, is conveniently effected by use of flaps.*

76. Flaps

Wing flaps exist in many different forms. They commonly
extend along the inner two-thirds or so of the span and are retracted

* The variable-pitch airscrew also provides, as one of its applications, additional
and powerful means for restricting landing runs.

FIG. 49. — LIFT CURVES AT Two SCALES FOR CLARK
YH AEROFOIL (ASPECT RATIO 6).

134

AERODYNAMICS

[CH.

into the wing section except when required for landing, slow flying,
or take-off. Size is specified by width expressed in terms of the
wing chord, and angle by the downward rotation from the with-
drawn position. Flaps should be located well aft. Several forms
move aft on opening, increasing the wing area ; in such cases
coefficients are reckoned on the original wing area.

In an early scheme for modifying wing sections during flight, the

(5)

FIG. 50.— WING

(1) Original form ; (2) Split flap;
(3) Split flap with displacement ;
U\ Original form slotted ;

(5) Split type slotted ;

(6) Split with displacement and

trailing edge slot.

8° 10° 12° 14° 16° 18°
INCIDENCE

FLAPS OF VARIOUS TYPES.

CL and CD at R » 1-7 x 106 :

(a) 20 per cent, flap type (6) at 30°.
(6) 20 per cent, flap type (2) at 45°.
(Partial span.)

IV]

AIRCRAFT IN STEADY FLIGHT

135

ailerons were rotated together to give maximum lift-drag ratio at
each speed before differential use for control, and depressed together
to assist landing. But the need to retain lateral control kept angles
far too small for the attainment of large lift and drag coefficients.

This original type, (1) of Fig. 50, could be employed at large
angles between the ailerons but is less effective than the ' split '
flap (2) of the same figure, which was invented (Dayton Wright) in
1921 and, like most ensuing types, leaves the upper surface of the

wing undisturbed. The split
flap was adopted generally in
1934 and enabled much larger
wing-loadings to be employed
without increase of landing
speeds. With its aid, the two
wings of a monoplane give as
much maximum lift as the
four wings of an unflapped
biplane, leading to greater
maximum speeds by reducing
skin friction and the parasitic
drag of external bracing (cf.
Article 69). Thus landing flaps
have so far been applied to
improve high-speed perform-
ance, and development tends
to continue on these lines,
but their reverse use is always
available to produce aero-
planes that will land especially
slowly.

The dotted curves (b) of
Figs. 50 and 51 relate to split
flaps and show (1) a large and
approximately constant in-
crease of CL at all flying inci-
dences, enabling high lift to
be realised without a high
and heavy undercarriage, (2)
little effect on stalling angle,
(3) little increase of CL beyond
the stall, and (4) a great in-
crease of drag at large flap

12° 14° 16°
INCIDENCE

18° 20°

FIG. 51. — FULL-SPAN FLAPS ON CLARK
YH AEROFOIL AT R = 3-9 x 10*.

(a) 40 per cent, flap type (3) at 45°.

(b) 10 per cent, flap type (2).at 90°,

136 AERODYNAMICS [CH.

angles. The severe drop of lift beyond the stall can be mitigated
by a so-called ' cut ' slot through the wing immediately in front
of the flap. Little is to be gained in lift as a rule by increas-
ing flap angles beyond 60°-70°, but larger angles may be used to
augment drag. Again, increase of width much beyond 20 per cent,
of the wing chord is seldom justifiable in view of extra weight and
operational difficulties.

With improved aerodromes, limitation of landing speeds is chiefly
important in connection with forced landings from low altitudes,
especially soon after take-off ; in normal circumstances weight is
much reduced by consumption of fuel before landing at the end of
a journey. Hence the very high wing-loadings frequently employed
for first-line aircraft present a more pressing problem in connection
with take-off than with landing. For this reason flaps are commonly
adjustable to give high lift and high drag for landing, with alter-
natively fairly high lift without undue drag for take-off. At (4) in
Fig. 50 is shown a type that is more useful for take-off than for
landing, a cut slot being fitted to the original form (1) and the hinge
being displaced backward and downward so that the slot remains
closed when the flap is not in use. At (6) is shown another form
capable of adjustment for the two purposes ; lift coefficients
exceeding 3 have been obtained with large flaps of this type extend-
ing over the full span.

Comparison with Air Brakes. — Before flaps came into general use,
air brakes of various forms were employed in addition to the
mechanical brakes fitted to undercarriage wheels. The extra drag
is readily seen to be small, unless the high-resistance area exposed
is large. Choose an aeroplane of 5000 Ib. weight with a flap as
given by Fig. 51 (b), but extending over the inner half of the span
and assumed to have one-half the effect. At 13° incidence the
basic wing shape gives r = 14, approximately, or, while the craft
is still air-borne just prior to landing, Dw = 5000/14 = 357 Ib.
With the flap, r = 7-8 and Dw = 641 Ib., an increase of 284 Ib.
This is independent of the speed, which depends upon the wing area
S. Fix this at 75 m.p.h. with flap. Since CL =1-56, S=5000/(l-564pF')
= 222 sq. ft. = Ic* for chord c and aspect ratio 7, so that c = 5-63 ft.
and the span = 39-4 ft. The area of the flap = 0-1 x 5*63 X \ X
39'4 = 1 1-1 sq. ft., a large area for so small a craft unless continuously
supported, as is possible with a flap. It is easily verified that this
area would not be reduced appreciably if the flap were separated
from the wing in the form of a simple air brake. For a long normal
plate, free along both edges, CD = 1-9, so that for the above area

IV]

AIRCRAFT IN STEADY FLIGHT

137

D = 0-95pF8 X 11-1 = 304 Ib. at 75 m.p.h., little greater than the
effective drag of 284 Ib. in position.

Spoilers. — These consist of long narrow strips, projected from the
forward part of the upper surface of the wings when the craft is close
to the ground and ready to land. Though themselves of small area,
they split the flow over the wing, causing the critical angle to occur
early, partly destroying lift and greatly increasing drag. Thus the
craft is let down to the ground quickly, giving wheel brakes oppor-
tunity to shorten landing run. But they do not permit the craft
to be brought in slowly.

Tabs. — Tabs may be regarded as very narrow flaps which are
fitted close to the trailing edges of control surfaces. Operated from
the cock-pit, servo tabs enable large control surfaces to be rotated
(in the opposite sense) with little effort, and trimming tabs alter
the zero positions of controls. Balance tabs are linked to control
surfaces to reduce operational effort in another way.

77. Power Curves

From preceding articles it appears that a maximum CL of 2-0 is
readily feasible with a large monoplane using a small flap. This
will be assumed. It also appears that minimum flying speed forms
a better gauge for wing area than landing speed.

Practical questions regarding aeroplane performance often lead
through equations (86)-(90) to cubic equations. Graphical presen-
tation avoids these. The process will be illustrated in the case of
an aeroplane weighing 10 tons, with reciprocating engines totalling
2000 b.h.p., and having a minimum flying speed of 60 m.p.h. Extra-
to-wing drag is assumed to be assessed at some high speed and to
decrease, in accordance with (90), to 110 Ib. at minimum speed.

TABLE VI

(V \2

Thrust

« (deg.)

L/D-r

Dw (Ib.)

£»B (Ib.)

£>, (Ib.)

(m.p.h.)

h.p.
required

- 1-0

0-14

13-0

1723

1571

3294

14-29

227

1990

- 0-7

0-16

15-1

1483

1375

2858

12-50

212

1620

-0-2

0-20

18-4

1217

1100

2317

10*00

190

1170

+ M

0-30

22-2

1009

733

1742

6-67

155

720

2-8

0-44

23-2

966

500

1466

4-65

128

500

7.7

0-80

18-1

1238

275

1513

2-50

383

10-4

1-00

15-0

1493

220

1713

2-00

388

13-3

1-20

12-6

1778

183

1961

1-67

77-5

405

16-7

1-40

10-8

2074

157

2231

1-43

426

18-3

1-48

10-0

2240

149

2389

1-35

69-5

444

19-2

1-30

6-5

3446

169

3615

1-54

74-5

717

A.D.— 5*

SPEED IN m p h

FIG. 52.

138 AERODYNAMICS [CH.

Given the first three columns
of Table VI, defining the
characteristics of the wings,
subsequent columns are com-
piled from equations (89) to
(92). All quantities relate to
low altitude. Columns 4-6
may be evaluated before the
speed, as tabulated, or after-
wards. The first column is of no
interest, except in locating the
wings on the body and in assess-
ing the CL available for landing. 19-2° is beyond the critical angle.
Drag of wings (flaps closed), of body, and total drag are plotted
against speed in Fig. 52. The variation of DB is parabolic within
the approximation contained in (90). Dw decreases by more than
50 per cent, while speed increases from 70 to 128 m.p.h. ; sub-
sequently it increases, at first slowly, but at high speeds quickly.
At 70 m.p.h. the body
contributes < 7 per cent,
to the total drag, but at
220 m.p.h. it contributes
48 per cent. These re-
sults, though special to
the present example, are
fairly typical of modern
craft of medium speed
and fine lines ; with low-
speed craft body drag
often appears in con-
siderably greater propor-
tion, mounting to high
values at a comparatively
early stage. An effect of
adding DB to Dw is to
decrease the speed for
minimum drag — from 128
m.p.h. to 112 m.p.h. in
the present example ;
greater parasitic resist-
ance would produce a
greater change.

1500

ciooo

500

100 150

SPEED (m.p.h)

200

FIG. 53.

IV] AIRCRAFT IN STEADY FLIGHT 139

The thrust h.p. required for horizontal flight is plotted as curve
(b) in Fig. 53. Curve (a) gives the thrust h.p. available from fixed-
pitch airscrews, less losses through extra drag of aircraft parts
within their slipstreams. The curve (a') relates to constant-speed
airscrews and assumes a continuously variable pitch. It will be
seen that the power available at intermediate speeds is greatly
reduced by a fixed pitch.

The dotted elbow in the h.p. required curve between 70 and 60
m.p.h. applies to flaps in use. There is only 100 h.p. to spare at 60
m.p.h. with a fixed pitch. It is by no means impossible for an
aeroplane to have wings equal to a lower horizontal speed than the
power units can manage (see p. 126).

78. Top Speed

The h.p/s available and required are equal at 211 m.p.h., the
maximum speed that can be reached under standard conditions in
straight horizontal flight ; if it is exceeded the craft must descend.
It occurs at a small negative incidence of the wings, viz. — 0-7°.
The lift-drag ratio of the wings is then 15 — far less than the maxi-
mum ; in craft of larger speed range the difference is greater.

Had only top speed been required, the first four rows of the Table
would have been sufficient. Complete curves have been obtained
for future reference. But to answer isolated questions it is economi-
cal to anticipate the result from inspection of the character of the
craft, and then to solve graphically through a short range of speed,
or, which comes to the same thing, of lift coefficient. This remark
also applies, of course, to the further analysis below.

79. Rate of Climb

It has been assumed, in preparing Fig. 53, that (88) or, more
generally, (84) can be satisfied at all flight speeds. Means for ensur-
ing this will be described later. Now assume flight to be taking
place at top speed and tail lift to be changed so as to satisfy (84) only
at some lower speed. If steady conditions are to result, speed must
decrease to an appropriate extent. If the engines are left at full
throttle, they will exert more power than is required for horizontal
flight and the craft will climb.

Alternatively, let the craft be flying at some speed lower than its
maximum with engines throttled, full power not being required.
Now let the engines be opened fully out without modifying tail lift.
Speed will remain practically unchanged from the value appropriate
to the tail lift, and consequently the work done per second in over-

AERODYNAMICS [CH.

Hence the additional power

coming drag will hardly change,
must produce climb.

A close approximation to the rate of climb is often obtained from
the assumption that speed is the same for given incidence, whether
climbing or flying horizontally. Then if Hf is the excess thrust h.p.
available at a given speed over and above that required for horizontal
flight at that speed, and v is the rate of climb in ft. per sec. :

v = 550 H,/W (94)

The reserve power is zero at maximum, and may be small at mini-
mum speed, but it attains a large maximum at some intermediate
speed with a craft of large speed range. The rate of climb is then a
maximum. In Fig. 53, curve (c) gives the reserve power for fixed-
pitch airscrews, attaining a maximum of 750 thrust h.p. at 128
m.p.h. when v = 550 X 750/22400 = 18-4 ft. per sec. Rate of
climb is expressed in ft. per min. and the maximum rate of climb of
the craft is 1104 ft. per min. The angle of climb is sin-1 (18-4/187)
= 5-6°, but this is not the maximum angle.

80. Climbing, Correction for Speed

With the simplifications already discussed, Fig. 54 shows the
forces acting on an aeroplane whose flight path is inclined upwards

at angle 6 to the horizon. Comparison will be made with horizontal
flight at the same angle of incidence of the wings. Climbing condi-
tions are distinguished by suffix c, and it is only assumed that weight,
together with the coefficients appropriate to the constant incidence,
remain unchanged. For steady climbing —

L* = Q, %?V*S = W cos 6 • ' • • (95)
Tc = 550 HC/VC = Dc + W sin 0 . . . (96)

It is seen that lift requires to be less than for horizontal flight at the
same incidence. This means a lower speed, the relation being —

(97)

IV] AIRCRAFT IN STEADY FLIGHT 141

and we have —

LJL «. DC/D » cos 6,
so that in climbing flight the thrust can be expressed as —

/ W \
Tc = Dcl 1 + — sin 6j = D cos 6(1 + ra tan 6)

whence it immediately follows that —

jf = (1 + ra tan 0) Vcos8 0 . . . (98)

ra being the lift-drag ratio of the complete aeroplane at the incidence
considered.

The approximate estimate of the preceding article can be written
in the form —

Hc = H + Wv/55Q
or —

jl = 1 + y: • y = 1 + ra sin 0.

Hence it is a conservative estimate. The errors in HC/H for 5°, 10°,
and 15° climbing angles are 0-4, 1-2, and 3-4 per cent., respectively,
for ra = 15.

A difficulty is sometimes felt with the implications of (98), in that
for a range of values of HC/H there are two values of 0, while for others
there is.no solution. Thus, writing HcjH = 2 and ra = 15, we find
0 = 4° or 89°, approximately. The small angle refers to flight of
the kind under discussion. At the large angle the craft would be
almost hovering, and would be of different form, of the type known
as a helicopter, and then practical difficulties in design would
prevent its taking up the corresponding horizontal flight. Thus,
certain second angles given by the equation lack practical interest.

Fig. 55 gives the form of (98) for various values of ra. Differen-
tiating —

[~ f sin 6(cos ° + r" sin 9) + fj'

This is a maximum when —

sin 0(cos 0 + rm sin 0) == f ra
or— J ra tana 0 + tan 0 — f ra = 0 ;

i.e. when— tan 0 = ~ *

It will be seen, from the figure or otherwise, that for all practical
aeroplanes 0 for maximum HC/H lies between 50° and 53°, and that

142

AERODYNAMICS

[Of.

ANGLE OP CUlVfB

FIG. 65. — NUMBERS ATTACHED TO CURVES GIVE OVERALL LIFT-DRAG RATIOS.

maximum HC/H = 0-618rfl + 0-5, approximately. It would be
feasible to construct a craft with sufficient power to exceed this
ratio. Incidence could not then be maintained and rectilinear climb
result ; if it were not decreased, the craft would begin a loop.
With reciprocating engines, the useful load of such a craft would be
very small, and one-half of the supposed power equipment consider-
ably exceeds the economic limit with present-day aeroplanes
intended for high-speed transport. On the other hand, the restric-
tion does not apply to military aeroplanes fitted with jet or rocket
propulsion. Gas turbines and jets, in course of development, will
enable large angles of climb to be attained by civil aircraft.

Referring to the example of Fig. 53, and taking Fig. 52 into
account, there are two speeds at which ra =« 15, viz. 97 and 132
m.p.h., the h.p. ratios being 2-65 and 2-35, and the climbing angles

IV] AIRCRAFT IN STEADY FLIGHT 143

6-3° and 5-1°, respectively. In a favourable case, variable-pitch
airscrews might be arranged to give 1300 effective thrust h.p., on the
present basis at the lower speed, when the h.p. ratio would be 3-4
and the angle of climb 8-9°. The direct importance of angle of
climb to civil aviation is largely in connection with take-off.
Reciprocating engines are sometimes boosted for short periods to
provide additional power for this purpose.

81. Effects of Altitude

So far, low altitude has been assumed. If altitude be increased,
air density diminishes (Article 14). Equation (86) then shows that
horizontal flight at a given CL, i.e. at a given a, can only continue on
increasing V, so that pF* remains constant. With this proviso,
Lwt Dw and DB are independent of altitude, ignoring modifications
in coefficients due to increased Aerodynamic scale. H, however,
increases as V, i.e. as '\fljja, where or is the density relative to that at
ground level.

Every point on a h.p. required curve (see Fig. 63) corresponds to
a particular incidence. Considering the effect of increased altitude
on any one such point, its ordinate and abscissa are both increased
in the ratio Vlja.

Had H^/c been plotted against F\Xa in Fig- 53> one h.p. required
curve would have sufficed for all altitudes. But on the basis of that
figure new curves for increasing altitudes can rapidly be derived.
Minimum h.p. will always occur at the same CL, and can be plotted

700

THRUST
H.P.

500

(b)

120 14O 16 O 160

V(m.p.h.)

FIG. 56.

separately against altitude if desired. Part of the curve of Fig. 53
is replotted for 20,000 ft. altitude in Fig. 56, curve (a).

Variation of performance with altitude depends more acutely,
however, on the power units. H.p. available decreases for altitudes
higher than that for which the engines are supercharged more
rapidly than the atmospheric pressure, Examination for a given

144

AERODYNAMICS

[CH.

engine and airscrew, or for a given type, involves technical questions
which will not be discussed at the present stage. But the rough
formula — thrust h.p. available oc a1*4 — is sufficiently representative,
for present purposes, of normally aspirated engines developing full
power at low altitude. The h.p. available curve of Fig. 53 has been
replotted on this assumption for 20,000 ft. altitude in Fig. 56, curve (b) .
It will be seen that, at the chosen high altitude, minimum flying
speed is increased to 129 m.p.h., top speed decreased to 159 m.p.h.,
and rate of climb decreased to 23 ft. per min. At approximately
20,500 ft. the curves have a common tangent at 145 m.p.h. ; the
craft will just fly horizontally at full power at this speed ; at any
other it must descend. The altitude at which the rate of climb is
zero is known as the absolute ceiling of the craft.

The reserve h.p. can be worked out by the above method for
various altitudes less than the absolute ceiling, and a curve giving
rate of climb against altitude follows. Fig. 57 gives this variation
without supercharging. Since a craft approaches its absolute
ceiling asymptotically, a ' service ' ceiling is introduced, defined by
the ^altitude at which the rate of climb falls to 100 ft. per min. This

is 18,300 ft. in the example.

Time of Climb.— The
time required by an aero-
plane to climb through a
given change of altitude is
clearly given by —

5000 KXOOO 15.000

ALTITUDE (FT)
FIG. 57.

2QOOO

tion of v with A, as illustrated in Fig. 57, so that

1 dh
v

where h denotes altitude,
and limits are inserted as
given. The time may be
determined by plotting the
reciprocal of v against h
and measuring the area
under the curve between
the limits prescribed.

For a normally aspirated
engine, however, there is
substantially a linear varia-

IV] AIRCRAFT IN STEADY FLIGHT H6

hi denoting the absolute ceiling and v0 the rate of climb at ground
level. If t is the time from ground level to altitude A', by integra-
tion—

L z.rfA==rlogi L77T ' - (")

82. Variation of Load

When the disposable load carried by an aeroplane is increased, the
abscissae and ordinates of points on the h.p. required curve at any
altitude both increase, the first because of the greater speed neces-
sary at any incidence, the second partly for the same reason and
also because Dw increases. Keeping incidence constant, we have
D oc Lac V* and H oc V* oc \/L*. Consequently —

enabling a h.p. required curve to be derived rapidly for any new total
weight. In the limit this curve will have a common tangent with
the h.p. available curve, when the absolute ceiling of the craft will
be at ground level. A near approach to this condition would be
dangerous, since the rate of climb would be very small.

The practical case arising is concerned with the maximum per-
missible total weight for a minimum value of the maximum rate of
climb, prescribed by local conditions or official regulations. The
method of solution will be obvious, A maximum weight is first
assumed from experience and parts of the h.p. required and available
curves are plotted, whence an estimate follows of the probable h.p.
available and incidence required in the limiting condition. An
equation can then be framed in W, having one term dependent on
the h.p. required for horizontal flight at the assumed incidence and
a second on the prescribed rate of climb. The solution can after-
wards be improved if need be.

83. Partial Engine Failure

Multi-engined aeroplanes must be designed to maintain altitude in
the event of one engine failing. The worst case is that of the twin-
engined craft with fixed-pitch airscrews. The h.p. available is then
cut by 50 per cent., whilst also the total drag is appreciably increased
by the head resistance of the useless airscrew. The drag coefficient
CD reckoned on projected blade area may be as great as 0*75, but is
usually somewhat less.

146 , AERODYNAMICS [CH.

Twin-engined layout has been assumed for the example of Fig. 53.
For moderate airscrew drag the h.p. required with one engine out of
action is represented by curve (e) . Curve (d) gives that available with
only one engine, and (/) the reserve h.p. with a fixed pitch. The
maximum reserve at low altitude is 107 h.p., indicating an absolute
ceiling of 5000 ft. An estimate on these lines of performance with
outboard engine failure must usually, however, be reduced owing
to the following consideration.

Airscrew thrust that is asymmetrical in plan leads to a yawing
moment on the craft, which is balanced by an equal moment arising
chiefly from a crosswind force on the rudder and fin. The resultant
of the two forces is inclined across the craft in plan, so that the
craft flies crabwise at a small angle of yaw. Total drag may be
appreciably greater in the yawed attitude.

It is often deduced from the above that three engines provide an
especially good layout. But it may be stated here that practical
cgnditions often dictate that there shall be an even number.

84. Straight Descent at Moderate Angles

If the flight path be inclined downward at 0 to the horizon, the
equations of Article 80 become (suffix e denoting descent) : —

!>• = CLip7/S = W cos 8

T. = 550H./F, = D, - W sin 0.

A descent with engines on is known as a power dive. Particular
solutions follow readily from power-curve analysis, but reciprocating
engines must be taken into account. Maximum permissible engine
revolutions are attained at a small angle of dive, and throttle must
be used for steeper angles until eventually the airscrews work as
powerful windmills, finally contributing a considerable fraction of
the whole drag. In very steep dives the above equations are in-
sufficient ; this case is considered in a later article.

There is a particular interest when T = 0, i.e. when the engines
are turning just sufficiently fast to prevent the airscrews from con-
tributing either thrust or drag, and when the angle of descent is
small. The case of engines off may be included, body drag being
increased on account of the airscrew blades. This form of flight is
known as gliding, and the equations give

r, = cot 0 (101)

6 is a minimum when ra is a maximum, corresponding to a certain
incidence for a particular craft, whence the speed of this flattest
glide follows. Thus, in the example of Fig. 52, airscrew thrust being

IV] AIRCRAFT IN STEADY FLIGHT 147

supposed zero, minimum 6 = tanT* (1420/22400) = 3'65°, CL «=0-50
approximately, and V, is only J per cent, less than the speed of
112 m.p.h. indicated in the figure for minimum drag in horizontal
flight. Two aspects of minimum gliding angle may be noticed.
Theoretically, it can be used to determine by observation in full-
scale experiment the maximum lift-drag ratio of a complete aero-
plane. But difficulties appear in practice. It is not easy to ensure
that T = 0, while also, as we shall find later, small upward trend of
the wind introduces large error. In case of complete engine failure
at a given altitude, minimum gliding angle determines the maximum
area from which the pilot can select suitable ground for a forced
landing.

Steeper descent is, of course, feasible, and there are then two inci-
dences from which to choose, corresponding to alternative speeds
for a given 6. It is desirable to be able to approach a confined
landing-ground at a steep angle and a low speed while avoiding very
large incidence in consideration of the comfort of passengers. For
this purpose ra must be low and CL large at a moderate incidence,
conditions which are excellently realised by using flaps.

84A. Induced-drag Method

The example of Article 77 may also be used to illustrate the utility
of the formula (v) of Article 69. Remembering that minimum
drag occurs when the induced drag D% is equal to the total parasitic
drag Z)P, and using the data deduced in the preceding article from
Table VI, each part of the total drag is equal to 710 Ib. at 112 m.p.h.
But for a given aeroplane in straight and level flight D< oc 1/F1 and
DP oc F2 to the present approximation, and hence at any speed
V m.p.h. —

This formula reproduces with the following errors various values
of DT listed in Table VI :—

V (m.p.h.) : 85 95 128 155 190 212
Error (%) : - 4£ - 1 + i - f - 1 - 4

Discrepancies are seen to be small through the major part of the
speed range. An isolated investigation of the present kind by no
means establishes the method, but similar examples combine to

148 AERODYNAMICS [CH.

verify that it can often be employed with fair accuracy except at
large or negative incidences.

Applications of the complete formula (v) of Article 69 to matters
such as those considered by other means in above articles will be
evident. For example, if the total weight of an aeroplane is in-
creased from Wl to W2, then the additional horse-power required
for the same speed is —

V _

550

DP remaining constant provided no great increase of incidence is
involved. The value of Dt at V with the weight equal to Wl may
be found as indicated above (methods of direct calculation are given
in a later chapter).

85. Effects of Wind

Aircraft speeds are always to be reckoned, of course, relative to
the wind. Ground speeds are obtained by adding vectorially the
wind velocity, provided that it has no vertical component. The
proviso is of great importance and seldom holds in practice. The
presence of an upward wind inclines the lift in horizontal flight for-
ward of the vertical, the aeroplane descending through the atmo-
sphere, and increased speed results for the same engine power
towards whatever point of the compass the aeroplane flies. One
method of calculating the effect follows from Article 59. Another
is as follows :

If v denote the upward component of the wind velocity, (101)
shows that an aeroplane will fly horizontally with zero thrust at an
incidence such that ra = V/v, approximately. The magnitude of v
required for sustentation can be deduced from the engine power
calculated as necessary at that incidence and speed for level flight
without upward wind, and any less upwind may be regarded as
leaving a corresponding proportion of the power available for
increase of speed. A greater upwind means that the aeroplane
would climb without airscrew thrust.

Powerless Gliders. — The foregoing principle is put to use in the
motorless glider. Gliders have essentially the same form as aero-
planes. But they are very lightly constructed, carry a minimum of
load, and have comparatively large wings, so that wing loading is
small and speed low. It is possible to realise high lift-drag ratios
in their case, especially in view of the absence of engine nacelles.

IV] AIRCRAFT IN STEADY FLIGHT 149

Consequently, a small value of v suffices for level, or even climbing,
flight. The latter is called soaring in the present case. Again, a
comparatively small head-wind enables them to hover. Rising
currents sufficient for soaring are found to the windward side of ris-
ing ground, and in many other circumstances, but they are especially
strong and extensive through cumulus cloud, and before the cold
air fronts of line-squalls ; with their aid great altitudes may be
gained, permitting cross-country glides exceeding 100 miles. Glid-
ing, by which intrepid pioneers explored the possibilities of flying
before the introduction of the light petrol engine, has now become
a recognised sport.

It will be appreciated that observations of top speed of engined
aircraft require correction for upwind if representative performance
figures are required. Another assumption in foregoing articles,
peculiarly affecting rate of climb, is that the horizontal wind remains
constant in respect of altitude. Suppose an aeroplane climbing
against a head-wind which increases (as is usual) with altitude.
With constant air speed, horizontal speed relative to the ground
becomes less with increasing altitude. The craft loses kinetic
energy, while its potential energy is increased by the wind at a like
rate. Thus the observed rate of climb is fictitiously great. Correc-
tion at altitude is easily made in this case, however, by repeating a
climb downwind and taking a mean of the observed rates. The
effect is of importance in the study^* of the take-off of aeroplanes,
and may greatly increase rate of climb near the ground.

86. Downwash

We proceed to study longitudinal balance, which has so far been
assumed. Although tail-less aircraft exist, longitudinal balance is
commonly secured by a tail plane fitted with elevators. These are
essentially affected by downwash from the wings, however, which
calls for prior consideration. Downwash is usually defined by the
angular deflection of the air in a downward direction from its undis-
turbed direction of flow, the craft being regarded as stationary, and
is denoted by e.

It is evident that wings lift by virtue of downward momentum
given at an appropriate rate to the air through which they fly. The
form this superposed air flow takes is complicated and its study is
deferred, but the downwash is, to a first approximation, constant

* Rolinson, A.R.C.R. & M., 1406, 1931.

150

AERODYNAMICS

[CH.

2C 3C

DISTANCE BEHIND TRAILING EDGE

(ce chord)

4-c

-2C

O C

DISTANCE ABOVE AEROFOIL

FIG. 58. — DOWNWASH BEHIND AN AEROFOIL IN A WIND TUNNEL, CL = 0-50.

(a) Numbers attached to curves give levels above trailing edge.

(b) Numbers attached to curves give distances behind aerofoil.

through the region occupied by a particular tail plane, if of small
span, and equal to that at its centre.

IV]

AIRCRAFT IN STEADY FLIGHT

151

Fig. 58* gives the downwash as measured in a 4-ft. enclosed-
section wind tunnel in the median plane behind a thin aerofoil
of 18-in. span set at 3° incidence (CL =0-50), showing variation

FIG. 59. — DOWNWASH 6 CHORDS BEHIND TRIPLANB IN WIND TUNNEL.
The curve marked * top wing * applies to a monoplane aerofoil of different
section (lift coefficient = C^) occupying the position of the top wing of the
triplane. The other two of the three lower curves are derived by reducing the top
wing curve in proportion to the known distribution of lift between the planes
of the triplane and displacing them to the levels of the appropriate planes.
The curve marked ' calculated ' is obtained by adding the ordinates of the three
lower curves and reducing the sum by the factor : Ci^/C^ = 0-76, Ci* being the mean
lift coefficient of the triplane.

(a) with distance downstream at various levels above the aerofoil,

(b) perpendicular to the span at various distances behind.

Well downstream, the distribution of e is little affected by minor

* The reader already acquainted with Aerodynamics or Hydrodynamics will at
once observe evidence in favour of the circulation theory of wing lift originally
advanced by Lanchester, developed by Praodtl and his colleagues, and now in
universal use. The observations recorded formed, indeed, some of the earliest
experimental corroborations advanced in support of the theory in this country.
(Piercy, Adv. Com. for Aeronautics, R. & M., 578, 1918.)

152 AERODYNAMICS [CH.

changes in aerofoil section, provided incidence is adjusted for con-
stant CL. As incidence (or, within limits, the section) changes,
aspect ratio remaining constant, the downwash at any fixed point
varies closely as CL through normal flying angles. Deviation from
this law occurs near the critical angle, and may also do so to a less
extent close to the incidence for no lift. Given the downwash dis-
tribution for a monoplane at a known CL, that for a biplane or multi-
plane wing system may be obtained by superposition, provided that
the CL of each member is known. Fig. 59* gives the results of super-
position by the proportionality law, together with direct measure-
ments made as a check.

Increase of e occurs locally in reduced velocity wakes. The wake
of a monoplane can sometimes be avoided by assigning a favourable
position to the tail plane, also desirable for other reasons. But it
will be seen that no reasonable position can be found that is removed
from the effects of downwash.

If at some wing incidence oc0, when the downwash at the tail plane
is e0, no lift is required from a tail plane of symmetrical section, it
must be set at the angle e0 to the undisturbed flow, i.e. at a0 — e0
to the wing. It is not as a rule fixed to the body of an aircraft at
the same incidence as the wings, and the difference is termed the tail
setting angle and denoted by a,. If wing incidence change, e will
change in the same direction, though at a less rate. Thus the effec-
tive change of incidence of a tail plane is less than the geometrical
change, and area has to be increased on this account.

87. Elevator Angle

To secure longitudinal equilibrium at wing incidence <x0 at a par-
ticular angle 6 of the flight path to the horizon, the tail plane and
elevators provide a particular moment about the C.G. of the craft,
balancing a contrary moment M0 that arises from other parts, par-
ticularly the wings. The tail plane is at incidence a' = a0 + a, — c0
to the local wind. If oc0 is inadvertently changed to a, a' becomes
a + a, — e and M0 changes to M , but the tail plane has at least suffi-
cient area to provide a force at its new incidence sufficient (taking
account of leverage about the C.G.) to overcome M — MQ and to
right the craft to a0. In symbols, dM/dtx, < — dMtjdy.t the minus
being introduced because the moments are of opposite sign. Stability
in regard to flight at a0 does not necessarily follow, but the above is
an important condition to that end.

If it is desired to change from a0 to a, the righting moment towards
* Piercy, Adv. Com, for Aev., R. & M., 634, 1919.

0°

-2*

-4°

-6°

-10°

-12°

100

125 150 175

SPEED (mph)

FIG. 60. — ELEVATOR CURVE.

200

223

IV] AIRCRAFT IN STEADY FLIGHT 153

a0 must be offset. This is achieved by adjusting the elevator angle
(measured between the centre-lines of the fixed part of the tail plane
and of the elevators) from Y)O, say, to YJ. By this means the tail lift,
positive or negative, is
reduced to precisely the
amount required for the
new compensating
moment. If now a
change without change
of Y), the complete tail
plane will right the aero-
plane back to a. It will
be seen that the role of
the elevators is to work
against the fixed part of
the tail plane when re-
quired.

Now a0, a correspond
to particular speeds of
flight at particular values
of 0. Thus the foregoing
argument may equally
well be expressed in terms of speed if 0 remain constant.

For small values of 0 we may ignore the difference between cos 0
and unity. In these circumstances we deduce that for a given craft
Y) determines the speed of flight, while it is the airscrews and vertical
wind which determine whether the craft shall fly level, or climb or
descend, at this speed.

Of course, with an unstable aeroplane, both a and 0 would be
indeterminate, and the maintenance of any average form of flight
would depend upon the skill of the pilot.

88. Example

Fig. 60 gives (inset) the lift coefficient Cu of a tail plane of aspect
ratio 3, free of downwash effects, through a restricted range of inci-
dence a' and elevator angle Y). The curves would be more openly
spaced with larger elevators. Increase of either a' or Y] results in
closer spacing, until eventually the tail plane stalls.

The 10-ton aeroplane of Article 73 in horizontal flight with flaps
closed is chosen to illustrate a usual method of investigating elevator
angle. Lengths are referred to a plane parallel to the wing chord
and passing through the C.G. of the craft. The C.P. travel in this

1 54 AERODYNAMICS [CH .

plane for the complete craft less tail-plane is given in the third
column of Table VII, the first two columns of which are copied from
Table VI. The C.G. is located at 0-3c behind the leading edge of the
wings of chord c, whence column 4 of the Table, % denoting the
distance of the C.P. measured in the plane in the upstream direction
from the C.G. No righting moment is required at 128 m.p.h., and
it is chosen to have the elevators neutral at this speed.

TABLE VII

a (deg.)

V (m.p.h.)

*0.*.

a' (deg.)

-n (deg.)

- 1-0

227

0-48

-0-18

-2-5

- 0-078

1-1

- 0-7

212

0-45

-0-15

- 2-3

— 0-074

0-9

-0-2

190

0-40

-0-10

-2-0

- 0-062

0-7

-f M

155

0-335

- 0-035

- 1-1

- 0-032

0-6

2-8

128

0-300

7-7

0-277

•f 0-023

+ 3-2

+ 0-056

- 3-6

10-4

0-272

0-028

4-9

0-086

- 5-6

13-3

77-5

0-269

0-031

6-8

0-114

- 7-9

18-3

69-5

0-266

0-034

10-0

0-154

- 12-2

The value 0-35, a theoretical result for a monoplane of aspect
ratio 6, is assumed for dtjdcf., a being the wing incidence, whence
follows the column of values of a' relative to the local stream. To
realise these, the tail setting angle a, must be — 0-8° ; for zero lift
occurs with the wings at a = — 3°, so that 128 m.p.h. applies to 5-8°
increase of incidence, when e = 0-35 x 5-8° = 2° = the incidence
of the tail plane to the direction of motion, whence a, = a' — a0 + e
= — 2-8° + 2° = — 0-8°.

St, the tail plane area including elevators, is taken as 13 per cent,
of that of the wings. Its C.P. is assumed to be fixed and distant
I = 2%c behind the C.G. of the craft measured in the plane. The
product of this length and St is called the tail volume. Further
assumptions made in order to avoid unnecessary detail are that the
tail plane avoids the wake of the wings ; that Ltt the tail lift, may
be neglected in comparison with Lw, the wing lift, so that LW~W\
and that moments of drags and of the airscrew thrust about the C.G.
may be ignored.

We then have, taking moments about the C.G, —

Lw . x cos a = Lt . / cos a ;
W x Lt I

or —

or-

'LI =

PF* c'
2W x c
^•"c'lSt

2Wx

(102)

IV] AIRCRAFT IN STEADY FLIGHT 155

if m is the tail volume. Values of W/pV* are calculated immediately
from Table VI, and column 6 of Table VII follows, or Cu may be
calculated directly by the relation : Cu = 3-08CL . x/c.

Finally, corresponding elevator angles are found from Fig. 60 by
interpolation, a' and Cu being now known. The same figure shows
T) plotted against speed (full-line curve). The curve is of typical
shape. The tail plane gives a righting moment against disturbance
at all speeds investigated, but the craft is very sensitive to longitu-
dinal control at speeds > 150 m.p.h., when \° movement of the
elevators suffices to add 60 m.p.h., although 5° movement is neces-
sary to decrease speed by the same amount. Control is still satis-
factory at 70 m.p.h., but is tending to become sluggish.

89. The student is recommended to work out further examples,
and should verify particularly that, although insufficient tail volume
must be avoided, another very important variable is the fore and
aft position of the C.G. This is nominally at the choice of the
designer, but a desired position cannot always be maintained under
varying conditions of loading ; for instance, 2 tons of fuel might be
consumed by the above craft during a non-stop flight of 1000 miles,
whilst structural difficulties might prevent balancing this bulk
precisely about a set C.G. Tail lift may be adjusted by trimming
tabs from the cockpit to compensate for shift of the C.G. during flight
or on changing disposable load. But this affects only in a secondary
way the problem before the designer, which is to determine what
displacement of the C.G. from its chosen position can be tolerated
for a given weight, having regard to the safety of the craft — here
represented by righting moment — comfort, and ease of control. The
broken line in Fig. 60 gives the result of moving the C.G. farther
back by 2 per cent, of the chord. Stability becomes neutral for
V > 150 m.p.h.

90. Nose Dive

The circumstances of an aeroplane in a very steep dive are excep-
tional. An interesting case is that in which the craft descends
steadily at fastest speed, engines off, a condition known as the
terminal nose dive. The flight path is then usually within 5° of the
vertical, so that the total drag is nearly equal to the weight. The
wings are nearly at the incidence for no lift, whence a first approxima-
tion to the high speed attained readily follows. But it is easily seen
that Lw, the wing lift, will not exactly vanish. For if it did so there
would remain a pitching moment due to the wings, which, together

156 AERODYNAMICS [CH.

with the moment of the body drag, now no longer negligible, must be
balanced by a tail moment. Lw is consequently required in general
to secure zero component force across the flight path.

The centre of pressure coefficient for the wings may be expressed
in the form A + B/CL, where A often lies between 0-22 and 0-25
and B between 0-02 and 0-10. For the very small lift coefficients
concerned, the C.P. of the wings may be near or even behind the tail
plane. Tail lift, Lt, may reach considerable values, and a practical
interest concerned with the strength of the structure centres in
determining its maximum value.

Let / be the leverage of Lit Dr the total drag, including the wind-
mill resistance of the airscrews, MT the total pitching moment,
excluding that of Lt. Neglecting body lift, we have —

whence —

(i)

£,=

(A,

or, dividing numerator and denominator by |pF»S —

lie

(103)

c being the wing chord, by (i). Calculations to determine the
maximum value proceed by assuming small increasing values for CL.
All the coefficients are expressed in terms of wing area and chord, but

it must be remembered
that they are composite,
and include the drag and
moment of drag of the
body.

91. Circling Flight

For an aeroplane of
weight W to fly uniformly
at speed F0 in a horizon-
tal circle of radius R, lift
and airscrew thrust must
balance, in addition to W
and the total drag Z)0, a

centrifugal force WVQ*/gR. With R large, crosswind force due to
flat yaw can be utilised for this purpose, the craft sideslipping, but

FIG. 61.

IV] AIRCRAFT IN STEADY FLIGHT 167

the normal course, necessary for smaller radii, is to bank the craft
at an angle <f> (Fig. 61), such that no sideslipping occurs. Then, if
LQ is the lift—

L0 cos <f> = W . . . (i)
L0 sin <f> = WV^/gR . . (ii)

so that— tan <f> = V^/gR. . . . (104)

But the equation— D0F0 = 550 H . . . (iii)

must also be satisfied. For the moment we assume the further
conditions regarding couples to be satisfied.

Comparing with straight horizontal flight at the same incidence
and altitude, since—

and- '=' = VsecU .... (105)

The abscissae of points on a h.p. required curve for a particular craft
in straight level flight are to be increased in the ratio l/\/cos <f> and
the ordinates in the cube of this ratio. New h.p. required curves are
thus immediately constructed for increasing values of <£, corre-
sponding to decreasing values of R at constant speeds.

Although h.p. required increases on turning into circling flight at
the same incidence, this is due, as the equations show, to the
necessary increase of speed. Comparing at constant speed, on the
other hand, gives —

Incidence must be increased for CLO > CL.

The increase of power required on changing from straight and
level to level circling flight at constant speed V is most readily
found from (v) of Article 69. Assuming that the increase of
incidence is not large, DP remains constant while Di increases in
the ratio (L^/W)2 = l/cosa $. Hence the additional power is —

VD
i

Di is the induced drag in straight and level flight at the speed
concerned and may be found as already indicated.

158 AERODYNAMICS [CH.

Rewriting (i) and substituting from (104) gives—

CLoip(gU tan <f>)S » W /cos <f> ;
or — R sin <f> = &/CLO,

k being a constant depending on wing loading and altitude. For a
given craft at constant altitude the equation may appear to suggest
R sin <f> to be a minimum when incidence increases on circling to the
stalling angle, and then R a minimum when <f> = 90°. But increase
of speed and drag prevent (iii) from being satisfied at a much smaller
angle of bank. Thus power-curve analysis decides minimum radius
of uniform turning subject to limitation of L0/W.

Of course, the direction of motion of an aeroplane can be reversed,
for instance, very quickly by using vertical bank and large incidence,
but the motion is unsteady, loss of altitude and speed taking place.

During circling, one wing tip is moving faster than the other and
a yawing moment arises, requiring to be balanced by the rudder.
Again, the tendency to greater lift of the faster wing must be com-
pensated by adjustment of the ailerons. Since also incidence is
increased, we see that all the controls are put into use.

92. Helical Descent

Direct descent preparatory to landing is conveniently effected by
flying down in a more or less vertical helix. Resolving along and
perpendicular to the helical flight path, angles of bank and descent
being $ and 0, respectively —

L cos ^ as W cos 0
L sin <{> = WV*/gR

T = D - W sin 6,

where R, the radius of curvaturfc of the path, exceeds the radius of
the helix in the ratio I/cos1 0. Compared with level circling, tan ^
is increased by the factor sec 0.

Notable effects occur when the wing incidence exceeds the critical
angle. Following a slight initial disturbance, the wings may then
produce a stable rolling motion about a longitudinal axis, known as
autorotation, and 0 and <f> may approach 70° or 80°, the radius of the
helix decreasing to a fraction of the span of the craft. This form of
flight is known as the spin, and in certain circumstances may in-
voluntarily result from stalling the wings at altitude. The matter is
investigated further in the following article.

93. Rolling and Autorotation

Let a monoplane of constant chord ct flying at speed V, receive an
angular velocity p about its longitudinal axis, so that the wing on

IV] AIRCRAFT IN STEADY FLIGHT 159

one side, beating downward, experiences a graded increase of inci-
dence, while on the other side incidence is decreased. Consider a
pair of wing-elements (Fig. 62), distant y on opposite sides of the
axis. The change of incidence Aoc at these positions amounts to
it yP/V> F°r a span of 2s the maximum values of this quantity are
± sp/V and occur at the wing tips. Provided the new incidence at

FIG. 62.

the downward-moving tip is considerably less than the critical angle,
we can regard dCL/d& as constant along the span to a first approxima-
tion. Then the lift coefficients of the elements at i y, originally the
same, are changed by the amount ± (dCJda) (yp/V), giving rise to a
couple :

T/i * dCL yp

pV* . c 8y . —^ .--- .y.
dcx. V

(a must be expressed in radians in this and similar expressions.)
Hence, neglecting the body, we have for the whole span the couple —

This is seen to be of large magnitude on inserting some practical numbers,
and evidently tends to damp out the rolling motion very quickly.

If, however, the monoplane is flying, prior to receiving the rolling
disturbance, at a low speed and a large incidence QCO, a small value of
sp suffices to invalidate the above method of calculation, and we
obtain a quite different result. Let us suppose the monoplane first
to be accidentally stalled, the constant incidence increasing to oi.

1 60 AERODYNAMICS [CH .

and then to receive a small p. For a lift curve of the type shown in
Figs. 49 or 63, downward-moving elements will now suffer decrease
of lift, while along the upward-moving wing some elements will
increase their lift. Considering again two elements distant ± y from
the axis, their changes of lift will now be different. Let ACL be the
whole difference of lift coefficient between them. The expression
for the rolling moment becomes —

which may be rewritten as :

018
016
04*
042
O40
0-8

10°

15° 20°
CL

25° 30°

-OO4

-o-osL

o-i

0-?

FIG. 63.

a form suitable for graphi-
cal integration. Plotting
ACL . yp/V against yp/V,
the area under the curve
(Fig. 63) as far as sp/V
is proportional to the
rolling moment at con-
stant p and V. As p in-
creases for a given craft
at a given speed, the
couple tends still further
to increase p at first,
but a limit is reached
when the couple is zero,
the integral vanishing as
is shown in the figure.
This corresponds to a par-
ticular angular velocity,
and the motion is evidently
stable, for any further
increase of p would pro-
duce a damping couple.

This striking result is
readily demonstrated in
a wind tunnel. An aero-
foil, or model of an aero-
plane, is mounted at a
suitably large incidence in
such a way as to be free

IV] AIRCRAFT IN STEADY FLIGHT 161

to rotate about an axis parallel to the wind. Slight disturbance
results in the model gathering angular velocity until a certain p is
reached, which it will maintain indefinitely. Timing this and
comparing with the value estimated as above usually shows good
agreement.

94. The Handley Page Slot

Recovery from a spin can usually be effected by decreasing inci-
dence, and nose diving to recover speed, but at low altitudes there is
no space for this manoeuvre. Thus it is important to retain lateral
control in case of inadvertent stalling near the ground. This insur-
ance is admirably effected by the Handley Page slot, a false nose to
the wing in front of the ailerons, which, on opening, considerably
delays the stall. We are not here concerned with the theory of the
working of the device, but Fig. 64 shows the effect in a particular
case, the slot extending the whole length of the aerofoil. Associated
with an increase of lift on opening the slot of a stalled wing there
occurs also a decrease of drag. This we shall find also to be a feature
of efficient lateral con-
trol, the wing that is
made to rise pushing
forward relative to the
other, so that the craft
turns in a direction
natural to the bank ;
the yawing moment
might easily have been
in the opposite direction,
necessitating compensa-
tion by use of the rudder.

As a result of this
brief investigation, we
note that delay of stall
is important, though not
of use in landing. The
ordinary flap is liable to
stall and induce auto-
rotation. To remove
this disadvantage, while
retaining high drag when
required, is the next
step in its development
A.D. — 6

FIG. 64. — EFFECT OF HANDLBY PAGE SLOT ON
LIFT CURVE.

182 AERODYNAMICS [CH. IV

and may be achieved by a slot system.* One form of slotted flap
has already been illustrated.

95. The Dihedral Angle

A damped roll by an aeroplane at normal incidence leaves the wings
banked, and, lift beinginclined away from the vertical, sideslip occurs,
the lower wing tip leading. Let the velocity of sideslip be v. The
wings may be regarded in the result as yawed at an angle sin~l (v/V),
the lower wing leading, and air passes the trailing edge of the
lower wing nearer to the body than it passes the leading edge.
If, when span is horizontal, each wing is inclined upward towards
the tip by a small angle p to the horizon, in the yaw equivalent to the
sideslip the incidence of the leading wing is increased approximately
by the amount $(v/V). The incidence of the trailing wing is simi-
larly decreased.

Considering a pair of elements of span 8y distant ± y from the
longitudinal axis, they give rise to a couple —

slC*

pF» . c8y . -~~ Aoc . y,
dv,

assuming incidence to be sufficiently small for the slope of the lift
curve to be constant. Hence the total rolling couple is —

(109)

The sense of this couple is clearly to right the aeroplane and stop
the sideslipping. Inserting practical numbers into the expression
shows the righting rolling moment to be powerful with the small
values of (3 used (cf. Fig. 61). The above estimate tends to be
excessive, in some cases by 30 per cent., owing to various neglected
factors, but a slight increase of p readily makes up for any such
deficiency.

The angle 2p is called the dihedral angle of the wings. As will be
anticipated, it becomes of great importance in the study of stability.
We shall note further here only that its magnitude requires adjusting
with some care ; too large a dihedral angle results in an unstable
motion of the craft.

* Cf. Nazir, Flight, Dec. 31st, 1936.

Chapter V
FUNDAMENTALS OF THE IRROTATIONAL FLOW

96. In Chapter II we found from experiment that the flow past
bodies shaped for low resistance comprises two dissimilar parts :
(a) a thin boundary layer, dominated by viscosity and merging into
the wake ; (b) an external motion, in which viscous effects are
scarcely measurable. In (b) occur the important pressure changes
which, transmitted through (a), account for part of the Aerodynamic
force on the body. Investigation will now be directed towards this
external flow. The fluid is assumed to be devoid of viscosity, so
that at any point the pressure acts equally in all directions. It
will later be proved that if an undisturbed stream of this inviscid
fluid is irrotational, it will remain irrotational in flowing past an
immersed body, since no tractions come into play which could
generate vorticity. Thus the total pressure head given by Ber-
noulli's equation remains constant throughout the flow. To take
account of the shape of immersed bodies, we must suppose that their
surfaces are closely approached, but not so closely as to enter the
boundary layer. This is tantamount to assuming that the boundary
layer is everywhere very thin and that no wake exists. In the limit
the fluid may be regarded as slipping with perfect ease over the sur-
faces of immersed bodies. The boundary condition for the idealised
fluid is, then, simply that the velocity component normal to the sur-
face vanishes. Attention is confined to two-dimensional conditions,
and compressibility of the fluid is neglected.

97. The Velocity-potential

A and B, Fig. 65 (a), are two points in a field of two-dimensional
irrotational flow parallel to the #jy-plane. For the present the region
is assumed to be occupied wholly by fluid. Join the points by any
curve, and let the velocity q make an angle a with the element 8s of
this curve. Write :

<£B — ^A = £ COS a ds . . . (1 JO)

J A
163

16* AERODYNAMICS [CH

This quantity will be shown
to have a unique value,
independent of the curve
drawn, as a consequence oi
the flow being irrotational.

Let ACB be another curve
joining the points, and con-
sider the line integral of the
tangential velocity com-
ponent once round the com-
plete circuit ABCA. The
area enclosed may, since it
does not include the section
of a body, be divided into a
large number of small fluid
parts by a fine network of
lines. The circulatory velo-
cities round the elements of
area so formed will cancel at
all common edges. There-
fore, the circulation round
the circuit equals, in the
end, the sum of the circula-
tions round all the elements
enclosed. Now it is assumed
that £ = 0 everywhere, so
that the element-circulations
all vanish separately ; there-
fore, the circulation round
ABCA is zero. Hence <f>B —
^A is the same whether

evaluated along AB or ACB, or along any curve joining the points.
Its value is therefore definite, and is called the change of velocity-
potential.

If A be fixed and B moved in such a manner that ^B — ^A remains
constant, B will trace out a line of constant ^, and conversely.
Thus the region of flow can be mapped out with contours of ^,
which are known as lines of equi-velocity-potential, or, shortly,
equipotentials. If zero value of 0 be assigned to one of these lines,
a numerical value follows for the velocity-potential along any
other line.
Now let A and B be adjacent points not on the same ^-contour.

FIG. 65.

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 165

If ty be the change of velocity-potential from A to B, we may calcu-
late it along AD, DB, Fig. 65 (6), finding-

s' = uSx + vfy.
But—

Hence —

«-»—

98. Physical Meaning of 0

Any incompressible flow having a definite velocity-potential could
be generated instantaneously from rest by a suitable system of
impulsive pressures. These might be applied from the surface of a
rigid body which is suddenly set in motion. At the point (xt y) in
the fluid let to be the impulsive pressure and u, v the velocity com-
ponents immediately after the impulse. An impulse is measured by
the change of momentum produced. Considering the element
8#8y, the change of momentum parallel to x is pw&*8y, while that
parallel toy is pv8#8y. The impulsive forces in these directions are,
A A

by Article 28, — ~ 8*8>y and — — 8#8y, respectively. Hence —

J ex oy

^-
ex

with a similar expression for v, or —

1 dw

"" p dx
I dm

Now comparing these equations with (111), we immediately find-
to = — p# + const. . . . (112)

The arbitrary constant refers to the general hydrostatic pressure,
and, if p is given its proper value, may be neglected while the
assumption of incompressible flow holds good.

This interpretation of <f> will be of particular interest later on, but
the following may be noted : (1) The equations for u and t; above
neglect all forces which are small compared with the very large force
acting for a short time which constitutes an impulse. Viscous

166 AERODYNAMICS [CH.

stresses would be in this category. Thus the equations certainly
apply momentarily to air. (2) Rotational motion has no velocity-
potential, and could neither be generated nor brought to rest by
impulsive pressures alone.

Irrotational flow is often called potential flow.

99. Since there is no flow along any part of a line of constant
velocity-potential, streamlines cross that line everywhere at right
angles. If the equi- velocity-potentials are closely mapped over a
field of irrotational flow, the system of curves that cut orthogonally
at all points of intersection will represent the streamlines. In
Article 38 the velocity components were related to the stream func-
tion fy by the following :

9d> 3ib

« = -—, v = — —"-. . . . (113)
dy ox '

Hence —

U(b CW C(b UW

dx dx dy dy "~~ '

which expresses the above result.

If the spacing of the curves accords with equal intervals of tf> and
fy, the resultant velocity q at any point, seen in Article 36 to be
inversely proportional to the distance apart of neighbouring stream-
lines, will also be inversely proportional to the distance apart of
neighbouring equipotentials. Mathematically, if Ss, 8n are elements
of length of adjacent streamlines and equi- velocity-potentials,
respectively,

ioo. Substituting from (111) in (61) gives for the equation of con-
tinuity for incompressible flow which is also irrotational —

9z«A a*0

_L J L 0

a*a ^ 3y«

This important equation, which occurs frequently in physics and
engineering, is known after Laplace. It is written for short —

VV=0 - • • • (115)
the symbol v* standing for d*/dx* + d*/dy*.

For irrotational incompressible flow Laplace's equation must also
be satisfied by the stream function. For substituting from (113) in
(65) gives —

o . . . . (no)

FUNDAMENTALS OF THE IRROTATIONAL FLOW

167

It follows that either the Alines or the Alines may be chosen as
streamlines, so that if the solution of one problem of irrotational
flow is known, the solution of a complementary problem also exists.
Every solution of Laplace's equation may be taken as representing
an irrotational motion. But to be of practical interest the solution
is additionally required to satisfy certain boundary conditions. The
straightforward calculation of </> and ^ in a complicated case where
the boundary conditions are prescribed is beyond the scope of this
book. But the solutions of several simple problems are easily found.
These are additive, because the equations involved are linear, and
hence more complicated motions can be built up. The final result
cannot as a rule be arranged exactly to comply with prescribed
conditions, but for many purposes it will give a sufficiently close
approximation.

1 01. Source

A source has no physical significance, but may be regarded as a
small circular area from which fluid flows out equally in all directions
in the #y-plane. Its
strength is defined by
the volume m of fluid,
per unit length per-
pendicular to the xy-
plane, sent out per
second. The stream-
lines are obviously
straight lines radiating
from the centre of the
source, and at radius r
the velocity q = m/2nr
and is wholly radial.
Suppose the source to
be situated at the origin
and choose Ox for the
streamline <J> = 0. The
flux across any curve
drawn from Ox will equal that across the arc of a circle of any radius
subtending the same angle 6 at the centre 0 ; this follows from
there being no flow across a radial line (Fig. 66). Therefore —

FIG. 66.

(H7)

AERODYNAMICS [CH.

Evaluating (110) along any radial streamline gives—

so that the equipotentials are concentric circles, as is otherwise
obvious. Choosing for 0 = 0 the circle of radius unity—

For equal intervals of fy tilt? streamlines are inclined to one another
at equal angles, and for equal intervals of <f> the logs (to base e) of
the radii will increase by a constant.

If the value of ^ (Fig. 66) is evaluated from the flux across AB, it
will differ by m from the value obtained from the curve ACD.
Evidently the value of fy for any streamline may be increased or
decreased by any multiple of m. The uncertainty is removed by
agreeing that 6 shall lie between 0 and 2n and <(;, consequently
between 0 and m. Other cases will occur where, as here, the value
of <]* or of <f> is unique, except for the addition of a ' cyclic ' constant.

102. Sink

Changing the sign of m in Article 101 makes the source into a sink,
a point or small circular area towards which fluid is flowing equally
in all radial directions in the #y-plane and at which it is supposed to
be disappearing.

A three-dimensional sink is a point or small sphere, the centre of
a symmetrical radial flow from all directions. The flow across all
surfaces completely surrounding the point will be the same. If this
is denoted by m, m is the strength of the sink, and the velocity at
radius r is m/4nr*.

Away from the immediate vicinity of the source or sink, where the
large velocities attained would make untenable the assumption of
incompressible flow, Bernoulli's equation applies in the simple form
P + \9f = const. It is easily found that the pressure drop varies
as 1/r1 for a two-dimensional, and as 1/r4 for a three-dimensional
source or sink.

Application to Experiment. — Measurements of drag are often
made in a stream of air which is slightly convergent in three dimen-
sions. A close approximation to the conditions is obtained by
assuming the body to be situated at a large distance r» from a sink.
If s denotes distance downstream measured from the position of the
body towards the sink —

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 169

Therefore —

^ (rf

W.^^
_ —

ds r0 —
But since /> + ip#s = const., differentiating —

_ _ _

ds ~~ p^ ' ds'
Hence —

dp _ 2P?»

1AA«'J

— — . .

ds r0 — s r0

approximately. Experiments with tunnels of parallel-walled type
show dp Ids oc ?a, giving rQ constant for a given tunnel. With tunnels
of the type illustrated in Fig. 26, r0 commonly amounts to 110 x
width of section.

An apparent drag arises on a body in a convergent stream. Since
this drag is inwardly directed, whether the flow is assumed to be
towards a sink or from a source, it can have nothing to do with Aero-
dynamic force ; it vanishes when r0 = oo, i.e. when the stream is
parallel. Article 59 (3) shows how measurements of Aerodynamic
force in a convergent Or divergent stream require correction for
pressure gradient.

103. Irrotational Circulation round a Circular Cylinder

Interchanging the meanings of <f> and <J> in Article 101, so that the
equi-velocity-potentials become the streamlines, we have the case of
fluid circulating irrotationaJly about a centre, and —

m . m

^ = -logr, *--e.

where m is now a constant whose meaning it is required to investi-
gate. The velocity q is perpendicular to r. Taken as positive in the
counter-clockwise sense, it is given by —

__ ty _ m
9 ^ ~~ Ijr ^ ~~2nr

and is constant if r is constant. Thus the circulation K round any
concentric circle is

K = 2-nr . q = — mt a const.

The circulation round all concentric circles being the same, it follows
that the circulation is the same round all circuits, of whatever shape,

A.D.— 6*

170 AERODYNAMICS [Cfl.

which may be drawn enclosing the centre, for any such circuit is
equivalent to one made up of arcs of concentric circles and of radial
elements, and along the latter there is no flow. To the value of <f>
for any radial line may be added a cyclic constant as for ty in
Article 101, but the convention there mentioned is again adopted.
On approaching the centre about which the fluid is circulating,
the velocity, which oc l/r, and the pressure drop, by Bernoulli's
equation, both become great. Apart from other considerations we
should expect to find eventually a hole in the fluid, because the
general hydrostatic pressure would be insufficient to support the loss
associated with the high velocity, a phenomenon known as cavita-
tion. The condition for the centre to be formed of fluid will be dis-
cussed later under vortices. For the present we assume the centre
to be isolated by a concentric circle, the trace of a circular cylinder,
of sufficient radius to prevent the velocity exceeding that which is
consistent with the assumption of incompressible flow when the fluid
is air. If the radius of the circle is a and this circle is chosen for
fy = 0, then for a greater radius r —

TjT

^ = ~~^l°Sa- . . . (120)

If a = 1, <|j = - (K/2n) log r.

A difficulty is sometimes experienced on a first reading in seeing
the necessity for irrotational circulation to have the above form.
An element of fluid circulating round the circular cylinder is in
equilibrium under its centrifugal force and the radial pressure
gradient. Thus, if V is its volume and r the radius of its path —

fv.f-v%=o.

r dr

From (120) q = — dfy/dr = K/2nr. Substitution leads to—

and on integrating —

Let p = P when r = oo. Then the const. = P and—

... (121)

v '

Now, because the flow is irrotational, this result must satisfy
Bernoulli's equation. When r = oo, q = 0, and we must have —

V] FUNDAMENTALS O* THE 1KROTATIONAL FLOW 171

On substitution this is seen to agree with the above result, explaining
the form determined for the circulation.

The motion investigated is an example of what is often called
cyclic flow, the cyclicity occurring in the value of <f>. Conversely, a
flow that is devoid of circulation is termed acyclic.

104. Combination of Source and Sink

The foregoing motions are supposed to be isolated. A source A
together with a sink B of equal strength provide an important com-
bined motion. Let A and B be situated on the #-axis at equal dis-
tances from the origin (Fig. 67). With Ay B as centres, draw arcs

FIG. 67.

PQt PR from any point P (x,y) to Ox, and let Ax, which is evidently
a streamline, be fy = 0. The flux across any line drawn from P to
Ax will equal the outward flow across PQ due to A, less the inward
flow across PR due to Bt or —

(122)

The streamline through P is (3 = const., i.e. the circular arc joining
A to B through P. Streamlines for half the field of flow are shown
in the figure. The equi-velocity-potentials are the orthogonal
systems of co-axial circles with A and B as limiting points. It will
be noted that (122) could have been obtained by simply adding
together the functions for a separate source and sink.

172 AERODYNAMICS [CH.

105. Doublet

Let A and J5 of the preceding article approach one another in-
definitely, so that the streamlines become the family of circles
touching the #-axis at the origin, as included in Fig. 71. Let m
increase as AB, which we will now write 8s, diminishes, so that in the
limit, when 8s becomes infinitely small and m infinitely great, the
product w8s remains finite and = (x, say. When (5 is small —

6 - 6' = tan (0 - 6') = . . ***** *„» x.
v ' x* + y* — ( JSs)1

and as 8s vanishes —

ty = — ^r- (6 - 6') = /- sin 0. . . (123)
Y 27c.8sv ' 2nr v '

A source and sink combined in this way is known as a doublet of
strength p.

106. The foregoing simple motions will now be combined with a
uniform stream of velocity U in the direction — Ox, i.e. — C7, whose
stream function is — Uy. The stream function of a resultant flow
is immediately obtained, as explained in Article 100, by adding
together the stream functions of its component parts, Laplace's
equation being linear. Details of the motion may be investigated
either analytically or mostly by graphical means, and the examples
given will illustrate both methods.

Flow over Symmetrically Faired Nose of Long Board or Plate. —
Consider a simple source at the origin added to the stream. The
stream function is —

<|, = -^ + £0. . . . (124)

Consider the streamline t]/ = 0. Either y = 0 or 6 = 2nUy/m.
Thus this streamline consists of the #-axis, together with the curve —

6 2U 2U . a

^ _. — y -- — r sin 0. . . (i)
TC w m

The curve is drawn in Fig. 68. It attains maximum values of y =
± m/2U, when 0 = ± TC and r = oo, i.e. at a large distance down-
stream. Where the curve crosses the #-axis, a stagnation point
occurs, for here the velocity due to the source cancels that due to the
oncoming stream, i.e. —

v]
giving-

FUNDAMENTALS OF THE IRROTATIONAL FLOW

178

Other streamlines are shown in the figure. The method of obtain*
ing these graphically is described in Article 107. Any of them may
be replaced wholly or in
part by a rigid boundary
without modifying the
others, because the fluid is
assumed to slip without
friction along a material
surface. Let us choose a
boundary in the position of
the curved part of ^ = 0,
and assume it to represent
the shaped contour of a
solid board or plate which
extends infinitely in the
direction — Ox and also
perpendicularly to the xy-

plane. The streamlines internal to the curve, of which four are
shown dotted, then cease to exist, and the source becomes an artifice
used to calculate the external streamlines, which give the inviscid
flow towards and over the nose^ of the board. The maximum
thickness of the board is seen to be 2n times the distance of the
stagnation point from the imaginary source.

Differentiating (124) :

FIG. 68.

u = — = — U
9y

m 30 __ , m cos 6

= — U 4- — . -
^ 2n r

msin 0
2n r

Hence, for the resultant velocity q at any point rt 6 —

wf mUcos 6

(ii)

Substitute in Bernoulli's equation :

p + i??1 « P +
where P is the undisturbed pressure, and obtain —

2 cos 6 —

(iii)

174

AERODYNAMICS

[CH.

enabling the pressure to be found at any point. But on the boundary,
i.e. over the surface of the board, m/27trU = sin 6/6 by (i) and —

<•«-•-»• • «

The pressure on the board equals P when 6 cot 6 = £, i.e. when
± 6 = 1-166 radians = 66-8°. From these points it increases
towards the extreme nose by £p£/2, while downstream it decreases
at first, but finally again approaches P.

We shall now investigate the drag D of the board. This will equal,
since skin friction is excluded, the total pressure exerted by the
shaped nose on the remainder. By Article 44 :

The pressure difference given by (iv) is plotted against y in Fig. 69,

and the area enclosed is seen to
y-m/2u vanish. Thus the drag is zero.

The result of zero drag is a
direct consequence of Bernoulli's
equation applying exactly through-
out the fluid, so that the fluid loses
no mechanical energy. But the
pressure-drag of the nose of a board
shaped in this way would be ex-
(f>-p)//pu* os pected to be small with air as fluid
FIG. 69. and with the real boundary con-

dition of absence of slip ; a drag

would exist, but this would approximate to the skin friction. In
the present example the pressures given by (iv) would, at least as
far along the board as the points of minimum pressure, differ little
from those which would be transmitted through the boundary layer
in experiment. For a board of finite length, if the section were suffi-
ciently long, the presence of a tail would not greatly modify the
pressures near the nose. Thus the distribution found approximates
to that existing over the fore-part of a symmetrical tail plane.

The %-axis beyond the stagnation point, together with the part of
the curve of fy = 0 to one side of the axis, might be chosen alter-
natively as boundary. Half the field of flow would then approxim-
ate to the flow of a uniform wind from a plain or sea over a cliff of the
section bounded by the curve. The application of this interesting
interpretation to motorless gliding is developed in the late Mr.
Glauert's Aerofoil and Airscrew Theory. Again, if the external
streamlines be ignored, we have the case of flow from a source

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 176

within a barrier, circumscribing the whole flow from the source.
Since the expanse of fluid is infinite in the complete problem, the
flow far downstream must be uniform and of velocity U. Hence,
the maximum width of the barrier is m/U as before.

107. Oval Cylinder

Assume a source and sink situated on the #-axis, the source at
x = + s, the sink at x = — s. Combining with uniform flow in the
direction Ox, we have —

fy = — Uy + — p. . . . (125)

This problem will be developed by the graphical method.

The streamlines of the combined source and sink and of the
uniform flow are known. Superpose these as shown in the lower
half of Fig. 70, attaching to each streamline its value of fy. The

FIG. 70. — ABOVE : STREAMLINES FOR POTENTIAL FLOW PAST OVAL
CYLINDER. BELOW : GRAPHICAL CONSTRUCTION.

closeness of packing of either set for equal intervals of <j* is open to
choice, but, together with the distance 2s, controls the final form of
the streamlines. At any point of intersection the value of ^ equals

176 AERODYNAMICS [CH.

the sum of the two values of 41 of the streamlines crossing at that
point. Draw a smooth curve through all points of intersection that
give in this way a constant resultant value of ^, and repeat the
process for different constant values. Then the curves obtained are
the resultant streamlines, shown in the upper half of the figure.

The streamline fy = 0 consists of the *-axis, excluding the length
2$, together with the oval curve shown. A set of streamlines
internal to this oval are ignored. Substituting a rigid boundary for
the oval, it becomes the contour of the section of a cylinder. The
condition determining the position of the front stagnation point
occurring on the axis Ox is that the sum of the velocities due to the
stream, source, and sink vanishes, i.e. if it is distant x0 from the
origin —

2n x0 — s
or —

,-±Vi

The condition fixes also a back stagnation point situated at an equal
distance on the other side of the origin.

The ratio of the length of the section to its maximum width across
the stream is known as the fineness ratio. The flow past cylinders
of different fineness ratios is obtained by varying the quantity m/Us.
Cylinders of elliptic section are treated in Articles 117 and 125.
The case of a cylinder of oval section moving broadside-on appears in
Article 149.

108. Circular Cylinder

A doublet of strength \i fixed at the origin with its axis (the line
joining the source to the sink) in the direction— 0%, together with
uniform streaming of velocity — [7, gives for the combined motion-
sin 6 _ / a \

(126)

Putting <J; « 0, we have for that streamline either y = 0 or
r = /y/ f g^jrA a const. = a, say. Thus a circle of radius a with

centre at the origin is a streamline. Let this be a boundary and
ignore the internal motion. Then the streamlines obtained from
(126), or by the graphical method of the preceding article, give the

FUNDAMENTALS OF THE IRROTATIONAL FLOW

177

flow past a long circular cylinder (Fig. 71), where are also shown the
streamlines for the doublet alone. (126) becomes—

- U (r- ~] si

sin 9

(127)

FIG. 71. — POTENTIAL FLOW PAST CIRCULAR CYLINDER.
Dotted : streamlines for doublet.

To obtain the velocity qa round the periphery, we have —

*-[- !L -

= 2U sin 8 .

(128)

giving stagnation points when 0=0 and TT, i.e. where the circle cuts
the #-axis. From Bernoulli's equation the difference between the
pressure at any point on the surface of the cylinder and P, the un-
disturbed pressure, is —

-&W=i (1-4 sin- 6). . (129)

The variation round the cylinder is plotted in Fig. 72, together with
some experimental measurements. There is fair agreement over

178

AERODYNAMICS

FIG. 72.

NORMAL PRESSURE
CYLINDER.

ROUND CIRCULAR

Hatched area includes experiments with R ranging
from 2 x 10* to 2 x 106. Original papers should
be consulted for variation in experimental data.

[CH.

the front part of the
cylinder which may be
extended at greater
Reynolds numbers, but
a real fluid breaks away.
From considerations of
symmetry it is apparent
at once that the drag for
irrotational flow is zero.
Thus the present theory
gives no help in calculat-
ing the drag of acylinder.
Nevertheless, we shall
find important uses for
the above results.

109. Circular Cylinder
with Circulation

On adding a counter-
clockwise circulation

K round the cylinder of the preceding article we obtain from
(120)-

2?r

(130)

The tangential velocity at r = a now comes to :

«• - IX1 + $ sin e + 2^L = 2U sin e

and for the pressure on the surface —

- P / . 2K sin 6

, 4

-. • (132)

The stagnation points no longer lie on a diameter, but approach
one another, being situated (if they remain on the surface of the
cylinder) at points given by qa = 0 or

sin 8 = —

(133)

y] FUNDAMENTALS OF THE IRROTATIONAL FLOW 179

an important result. When

K = 4naU, sin 6 = — 1

and they coincide on the

bottom of the cylinder. If

K/U be further increased,

(133) does not apply ; they

still coincide on the axis of

yt but occur in the fluid.

The streamlines in the latter

case are shown in Fig. 73,

S being the stagnation point.

The fluid between the cylin-
der and the loop encircling

it from S circulates con-
tinuously round the cylinder,

failing to pass downstream.

value of K/U are given in Fig. 74.

It is again obvious from symmetry that the drag is zero. But the

pressure is less on the upper
half of the section than it is
on the lower half. It is im-
mediately found from (132),
for example, that if pi be
pressure at the top of the
cylinder and p^ that at the
bottom :

FIG. 73. — POTENTIAL FLOW PAST CIRCULAR
CYLINDER WITH STRONG CIRCULATION.

The streamlines for a much smaller

P. -

FIG. 74. — POTENTIAL FLOW PAST CIRCULAR
CYLINDER WITH WEAK CIRCULATION

denoting by q' the velocity at

r = a of the circulation alone. Consequently, a lift L arises. To
find this we note that the lift 8L of an element 8s (= a . 80, Fig.
75) of the contour is — (p — P) a8Q sin 8,
so that on substituting forp — P from
(132) and integrating with regard to 6
between the limits 0 and 2;r, all integrals
except that derived from the third term
of the R.H.S. of the equation vanish,
since they contain sin 6 to an odd power.
Hence :

r pt/K f«- . § .
L = sin8 0

7C J 0

(134)

FIG. 76.

180 AERODYNAMICS [CH.

This gives a lift coefficient :

-- • • (135)

The above result is of great importance. The lift is independent of
the size of the cylinder. A circulation can be generated by rotating
a real circular cylinder in air, when, if it also moves as a whole, a lift
of this kind appears, although the flow is not wholly irrotational.
The principle finds practical expression in Flettner's sailless ship and
in many ball games. We shall find that a cylinder of wing-shaped
section moving through a viscous fluid has the property of generating
a circulation by other means, and, with the help of an analytical
process to be explained later, we shall be able to calculate the lift of
wings with good accuracy from the basis provided by the foregoing
results.

no. The Potential Function

The complex function <f> -f fy, where i denotes V(~~ 1)* *s called
the potential function of the irrotational flow. Let us equate it to
any analytical function of the complex variable x + iyt say/(# + iy),
so that—

# + **=/(* + *». . . (i)

Then we have —

and —

9y 3y ~~ •"

Hence, equating real and imaginary parts-

which, Article 100, are the relations requiring to be satisfied for
irrotational flow. Therefore, any assumption made in accordance
with (i) leads to an irrotational motion. For shortness it is usual to
write :

z =x + iy,
w = <f> + ify.

The function of z,f(z), can always be separated into real and imagin-
ary parts. Then from (i) we immediately obtain <f> and ^, which are
real functions of x and y. It will be noticed, however, that the
method can be applied only to two-dimensional problems.

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 181

It is shown in the theory of functions that if w ==/(*) —

dw d(<p -j- ^T) ®*r • ^T v(p . dit>

dz ~~~ d(x + iy) dx dx dx dy'
Thus, from (111) :

-=u-w .... (136)

in. As a first example it will be shown that —

»=/(*) = (4 +iB)z . . . (137)

where A and B are real constants, covers all cases of uniform motion.
We have —

<£ -f- ty = A(x + iy) + iB(x + iy)
= Ax — By + i(Bx + Ay).
Equating real and imaginary parts —
^ = Ax — By
fy = Bx + Ay
and —

U=Zfa~A

v =— = — B.

Both velocity components are constant for chosen values of A and B
and the flow is therefore uniform. If B = 0 the constant of (137) is
wholly real and the flow is in the direction Ox with velocity U = A.
If A = 0, the flow is parallel to 0jy and of velocity V = — B.
Generally, the flow is inclined to the #-axis by the angle tan""1
(— B/A), the velocity being equal to VM1 + B9).

1 12. It is often convenient to express the complex variable z in the
polar form :

z = x + iy = f(cos 0 + i sin 0).
Remembering that —

cos 0 =

and —

sin 0 =C- — ^ —
2i

we note that —

cos 0 + * sin 0 = e**
cos 0 — i sin 0 = 0~*° .

182 AERODYNAMICS [CH.

If we have x in the form % + iy, it can always be obtained in the
form —

z = re®.

For, writing out both sides and equating real and imaginary parts,
we find x = r cos 0, y = r sin 6, so that r = V(*2 + 3/2)> of which the
positive root is taken, and these equations give a unique value of 0
between 0 and 2n. r is called the modulus of z and is written mod z
or |z| ; 6 is called the argument of z.

The complex co-ordinate z can be represented geometrically.
For the operator — 1 applied to 0% changes it to — Ox, i.e. turns

it through the angle TC. Since — 1 =
i x i, the operator i turns a length
through a right angle. Hence, to plot
the point % + iyt x is measured along
Ox and the increment y is measured at
right angles thereto (Fig. 76). If P is
the point represented by z, it will be
seen that OP = r and tan""1 (y/x) = 0.

O X Thus z represents the vector OP, its

FlG 76 length being \z\ and the angle it makes

with Ox being 0.

113. With this brief note on the complex variable, we proceed to
consider the function —

w =/(*) = Az + A/z. . . . (138)

By Article 111 the first term on the R.H.S. represents steady
streaming at velocity U = A, and to this is added a second motion.
The combination may be written in the polar form :

w=A(re* + -«"*)
and we have —

^ -f ity = AY (cos 0 + i sin 0) -f — (cos 0 — i sin 0).

Equating real and imaginary parts —

$ =A(r + l/r) cos 0
4* =A(r — l/r) sin 0.

Comparing with (127) or by considering the form of the streamline
fy as 0, we find that (138) gives the flow at velocity A past a circular
cylinder of unit radius. It may be noted also that w = A/z repre-
sents a doublet at the origin, as may be verified independently.

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 183

H3A. Formulae for Velocity

The velocity q at the general point in a two-dimensional irrota-
tional flow can be expressed in various ways, of which the most
useful are the following :
(a) Directly from (111), since q* = w1 -f v*,

(b) Denoting the components of q along and perpendicular to the
radius r from the origin by u' and v', respectively,

- -^5 and v' = — ^r>
r 30 3r

whence — q = («'* + t/1

= («' + v')1'2. Hence— q =

(iii)

For the potential function of the preceding article, for instance,
(iii) gives, since —
dw
~dz

1/2

] = A \ I — — (cos 26 — t sin 26) >• ,

W I y1 J

FIG. 70A.

1/2

As another example consider the func-
tion— c

w —~~znt
n

which yields a variety of irrotational
motions on ascribing different values
to n, choosing straight streamlines
through the origin as boundaries, and
interchanging if need be the meanings
of <f> and ty ; e.g. a doublet with n =
— 1, the streamlines (consisting of
rectangular hyperbolas) in the vicinity
of a stagnation point with n — 2,
Fig. 76A, etc. For all these,

9= \dwjdzl - |C*"-J | - O"-1,
i.e. the velocity is constant at a given
radius from the origin.

1 84 AERODYNAMICS [CH .

114. It will be convenient to have the potential function for a
cylinder of unit radius with circulation K in a stream of velocity— U.
Let—

w = - * ~ log z. . . . (139)

2?u

Now the Napierian logarithm of x + iy ( = re?9) is log r + *6. Hence —

K K

* + .-*--.-logr + -e

and —

,„«(,, t — £h«r

so that (139) gives circulation with strength K round the origin, and
the expression for ^ is unchanged if the circulation is round a circle
of unit radius.

Hence, for this circulation combined with translation we have,
from Article 113 —

US. Instead of w being expressed as a function of z, we may have
z as a function of w.

Consider :

z = c cosh w . . . . (141)
Writing out —

x -f iy = c cosh ($ + *^)

= c (cosh <£ cosh *4> + sinh ^ sinh i^)
= c [cosh <£ cos fy + sinh <£ (i sin <{;)].

Equating real and imaginary parts —

# = c cosh <£ cos i|>

y = c sinh ^ sin i|; . . . (i)

Square and add to eliminate <J>, obtaining —

x* v1

-j = cos1 d/ -f~ sin1 ^ = 1 (ii)

c2 cosh2 ^ c1 sinh1 ^

or square and subtract to eliminate <£, finding alternatively —

^ Z — = cosh1 <k — sinh1 <A = 1. (iii)

c1 cos1 1|/ c1 sin1 <j>

Putting <^ = a series of constants in (ii) gives a family of confocal
ellipses (Fig. 77), the foci being at % = ± c> y = 0. Choosing ^ as
the stream function, any one of these ellipses may be taken as

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 185

FIG. 77. — IRROTATIONAL CIRCULATION ROUND PLATE.

boundary, and we then have the streamlines for irrotational circula-
tion round a cylinder of elliptic section. The line joining the foci
may be taken as boundary, when the ellipses become the streamlines
for circulation round a flat plate. It is readily seen by plotting or
calculation that the velocity and the pressure reduction both become
very large as the edges of the plate are closely approached, and we
shall frequently have to remark on artificiality on this score. It will
be noted that, at a large distance from the plate or elliptic cylinder,
the streamlines become the same as for circulation round a circular
cylinder.

Putting <|/ = const, in (iii) gives a family of hyperbolas having the
same foci. These are everywhere orthogonal to the ellipses, and con-
stitute the equipotentials of the circulation. If, however, they be
interpreted as the streamlines, so that the ellipses become the equi-
potentials, we shall have the case of fluid flowing through the whole
or part of the #-axis between ± c. Choosing two hyperbolas equi-
distant from the ^-axis as boundaries, we at once have the stream-
lines for flow through a long two-dimensional nozzle (Fig. 78).
According to potential flow theory the nozzle may be made as sharp
as we please, but a real flow breaks away from the divergent

186

AERODYNAMICS

FIG. 78. — POTENTIAL FLOW THROUGH HYPERBOLIC
CHANNEL.

[CH.

walls, the flow ceasing
to fill the channel, if
the divergence is other
than small. With this
restriction, the three-
dimensional analogue
is applied in the
design of high-speed
wind tunnels. Re-
covery of pressure
energy at the outlet
from the kinetic energy
generated at the
throat leads to higher
efficiency (Article 51),
resulting in greater
speed at the throat
for a given expendi-
ture of power, than if the tunnel were parallel-walled. The idea is of
ancient origin. A cylindrical tunnel is often fitted with a divergent
outlet only. Where smooth flow and high efficiency are urgent, it
is advisable to shape, if possible, the divergent wall with some care.
The inlet part is of less importance, and is often made of quite
different form for other reasons.

The complete nozzle is known as a venturi or venturi-tube, and,
in its three-dimensional form, has many practical applications. A
pressure reduction obviously occurs at the throat, and, if it is known
in a given instance to what extent the space between the walls is
filled with continuous flow, this reduction follows at once from
Bernoulli's equation. When the venturi forms part of a pipe-line
conveying liquid, the convergent inlet is forced to run full. But if
the venturi is short and is exposed in a stream, free to flow round it,
little fluid may pass through, so that it by no means runs full.*
Nevertheless, a pressure reduction still exists which can be used
(after calibration) to measure velocity (Article 33), or again to supply
power. Application to aircraft in the latter connection is associated
with poor efficiency.

1 1 6. Motion of a Cylinder through Fluid

So far the immersed body has been assumed to be held in a stream.
Sometimes it is desirable to consider the body as moving, the fluid

* Piercy and Mines, loc. cit., p. 44.

FUNDAMENTALS. OF THE IRROTATIONAL FLOW

187

being stationary at large distances away. If the solution be known
in the former case, that in the latter may readily be deduced by the
superposition of an additional stream function as already explained.
But the direct solution may be simpler and a method for this will
now be described.

The boundary condition can now be stated as follows : the resolved
parts of the velocities of an
element of the contour of
the body and of the adja-
cent fluid, along a normal
drawn from the element,
must be the same.

The contour shown in
Fig. 79 is that of the cross-
section of any cylinder
supposed to move at uni-
form velocity U in the
direction Ox. Distance

_
O X

FIG. 79.

measured round the curve

in the direction of 0, increasing as shown, is denoted by s. Con-
sidering a small element Ss, the velocity component of the cylinder
along the normal from it is f/.cos 0, while the velocity component
of the fluid there in the same direction is v . sin 0 + « • cos 0.
Therefore —

v sin 0 + u cos 0 = U cos 0.
Substituting for u and v, and from the figure —

ds/ r dy ' ds ds9

noticing that x is decreasing at the element in the figure as s is
increasing. Hence :

— dx + TT~ dy = dk = Udy.
ox dy

Finally, integrating round the boundary —

4, = Uy + const (142)

Any form of ty satisfying Laplace's equation (Article 100) gives
from this expression a family of curves any one of which may become
a boundary and, moved in the direction Ox, will give the path-lines
<|t = constant. A similar expression is obtained for motion parallel
to the y-axis. Superposition of motions parallel to x and y enables
path-lines to be obtained when the cylinder moves with its section

188 AERODYNAMICS [CH.

inclined. The streamlines for the body at rest are immediately
found by the addition of an appropriate stream function affecting
the fluid as a whole.

The method was employed by Rankine * to find mathematical
shapes for ship lines. It is tentative or inverse in the sense that the
form selected for ^ (and there is an infinite number) may well lead
to a possible variety of shapes for the boundary, none of which has
any bearing on Aerodynamics. The following classical example has
a particular interest, and should be studied carefully, as we shall use
it later on as a key to a difficult problem of the greatest practical
importance in our subject.

117. Elliptic Cylinder and Plate in Motion
Assume for the potential function the form —

w = — Ac-(* + i*. . . (i)
where A is a real constant. We have —

^ -f fy = — Ac~* (cos Y) — t sin YJ)
and on separation of real and imaginary parts —

<J> =<4*~f sin Y) . . . (ii)

The co-ordinates 5, YJ, called elliptic co-ordinates, are related to
xt y in the same way as <f>, fy were in Article 115, i.e. —

z = c cosh (? + f>j) ;
so that —

x = c cosh £ cos Y],

y = c sinh £ sin YJ. . . (iii)

As in that article, we find that 5 = const. = £0, say, is the ellipse :

which we shall write for short —

so that —

a = c cosh £0, b = c sinh 5o . . (iv)
are its semi-axes.

Now putting (ii), with 5 = So so as to represent the boundary, in
(142) gives, making use of the second formula of (iii) —

Ae~*9 sin YJ = Uc sinh 5o sin YJ + const. ;

* Phil. Trem$. Roy. Soc., 1864.

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 189

and since this must be satisfied for all values of >), we have —

the const. = 0,

A = Uce*9 sinh 5o-

Hence, in order that (i) should represent the case of the ellipse

ty = Uce**~* sinh ^o sin YJ . . (v)
This result can be simplified, because, using (iv) —

t b

cr* sinh £o = bef* = b (cosh £0 + sinh £e) = - (a -f b) ;

and —

c1 SB a1 — ft1.
Thus, finally —

' ^ *-*ci«v» n/lQ\

— £ . c * sin Y) . . (I4o)
«• — o

and —

= — Ub A/ - _ . c~*
r a — 0

cos

These expressions are for motion parallel to the major axis.
Corresponding results are similarly obtained for motion in the direc-
tion of the minor axis. They come to —

cos 73, . . (144)

^ = - Va /\A±4. e~s sin Y,.
v a — 6

The solution applies to all confocal ellipses, and so includes the
case of a plate of chord 2c (cf . Article 1 15) moving broadside on. In
this important case 6=0 and a = c, and the last formulae become —

^ as — Vc t~* cos >j

<£ = — Vc e~* sin >).... (145)

The path-lines are shown in Fig. 80 for downward motion of the plate.

If the cylinder or plate has
components of velocity U and /^ ^N^ \

V, Le. if the line joining the foci,
or the plane of the plate, be
inclined to the direction of mo-
tion, the new stream function

is written down immediately by ^ rtxv ^ _

_. A Al_ . J J FIG. 80.-— PATH-LINES FOR PLATE IN
Superposition. But the Stream- BROADSJDB-ON MOTION.

100

AERODYNAMICS

[CH.

FIG. 81.

lines are more illustrative than the path-lines, and for these the
additional stream function Vx — Uy, where x and y are given by
(iii), must be superposed. In this way the streamlines for any angle
of incidence a can be plotted ; they are shown for a flat plate at

a = 45° in Fig. 81, the
oncoming stream being
supposed horizontal.
Another treatment is
given in Article 124,
where further details
are obtained.

From symmetry,
there is no component
of force in any direction
on the plate or cylin-
der, whatever its in-
cidence, although, if it
be inclined, a couple

exists tending to produce broadside-on motion. This further
instance of absence of force in steady motion is reviewed in the next
article. A circulation might be added from Article 115, and a lift
or transverse force would result ; this will appear as a special case
of a more general investigation in the next chapter.

1 1 8. It has been remarked several times that the only Aero-
dynamic force arising on a body in steady potential flow is that due
to the superposition of circulation and translation, and is a transverse
force. Absence of drag is especially striking, perhaps, in the example
just considered of a flat plate moving at right angles to its plane,
when the drag coefficient CD has, in experimental fact, a value equal
to 2, representing a particularly large force. It may be remarked
in this case that high velocities are built up towards the edges which
would invalidate the assumption of incompressible flow with air as
fluid, and even with a liquid such pressure reductions would occur
before the edges were reached as could not be supported by the
general hydrostatic pressure. To avoid these objections, which
would clearly prevent the flow from running smoothly to the back of
the plate, we might round the edges, as, for instance, by substituting
an elliptic cylinder. But the flow would still break away, as was
seen to occur even with the circular cylinder. The subject of drag
is complicated, and is postponed until later chapters. But it should
not be inferred that failure to indicate drag prevents the foregoing
theory from being of practical use. The methods discussed will

V] FUNDAMENTALS OF TliE IRROTATIONAL FLOW 191

often suffice to calculate approximately the streamlines and velocity
and pressure distribution over the fore-parts of bodies. They could
readily be developed to a more effective stage. But this will be left
to a reading of original papers, for, from the foregoing theory, we can
proceed directly to a very powerful process of solution that readily
gives essentially practical forms of potential flow. This is treated in
the next chapter.

119. Acceleration from Rest

There is one circumstance, however, in which potential flow yields
a drag, viz. during the time of its generation.

Consider a body at rest in an infinite bulk of stationary fluid. Its
weight will be assumed to be balanced by its buoyancy or by
mechanical means. Let it be given an impulse in any direction,
i.e. let an indefinitely large force act upon it for an indefinitely short
time T, being withdrawn at the end of T. The impulse is measured
by the momentum produced. In vacuo, the impulse would be given
by the momentum acquired by the body. But part of the impulse
is absorbed in generating momentum in the fluid. This increment
alone concerns us, and we shall denote it by /.

Now, regarding the flow generated in the fluid by the motion of the
body under 7, it can be proved that if the acyclic potential flow is
known for that body, then the flow actually set up will be of that
known form. We need not follow out the theoretical argument,
because, as will be described in more detail later on, the result can be
verified by experiment with a real fluid, whose viscosity requires
appreciable time to take effect and so modify the flow. Thus, the
present investigation relates to the initial motion of air, provided
generation from rest is almost instantaneous.

Assuming that the body is of such a shape that the solution for
irrotational flow exists, we know from Article 98 the distribution over
its surface of the impulsive pressure which generates the flow. The
pressure acts normally to the surface of the body at all points, and,
on integration over the body (cf. Article 44), will give a resultant
force which must exactly balance that part X of the external force
applied which is not absorbed in producing momentum in the body,
and we have —

x = Tt <146>

At the end of the short interval of time T, when the impulsive force
is removed, the motiop of the body and the fluid becomes steady, if

192 AERODYNAMICS [CH.

the latter is inviscid, and the pressures round the body indicate zero
resistance, as we have seen. With air, viscosity produces friction
and soon modifies the flow, leading to pressure drag, as well as skin
friction, as also we have noted. But the important result now
obtained is that during instantaneous generation of flow from rest,
whether the fluid possess viscosity or be conceived to be destitute of
this property, a force must be applied whose magnitude, direction,
and point of application can be calculated in suitable circumstances.
It is clear that a linear impulse would be insufficient to generate
some motions, and that an impulsive ' wrench ' would sometimes be
required. But the moment of the impulse may be found similarly.

Now during T the impulse / does work on the fluid, evidenced by
the appearance within the fluid of kinetic energy. Denote the
kinetic energy at the end of T by E. It is given by—

E^toJJfdxdy . . . (147)

in the two-dimensional case, if E be reckoned for unit depth per-
pendicular to the ay-plane, q denote the velocity at any point, and
the integration extend over the whole of the ay-plane that is not
occupied by the section of the body.

The above integration is, in general, difficult to carry out. But
E must equal the work done by the impulse during T. Now a
familiar theorem of Dynamics proves that the work done by a
system of impulses operating from rest is equal to the sum of the
products of each impulse and half the final velocity of its point of
application. This theorem may be applied to the finite continuous
distribution of impulse which we have to consider. If 8n be an
element of the normal drawn into the fluid from an element 8s of the
contour of the body, the final velocity at 8s is dfydn, while the ' im-
pulse pressure ' at 8s is, by Article 98, — p«£. Hence—

S ' ' ' (W8)
where the integration is to extend round the contour of the body.
Thus the kinetic energy at the end of T is at once calculated if <f> be
known.

With an inviscid fluid, E remains constant after T. The flow
might, however, provided it is irrotational, be brought instantane-
ously to rest by the application of a reverse impulsive wrench, which
would do work in destroying the kinetic energy of the fluid. Thus
the work done by an impulse is equal to the (positive or negative)
increment of the kinetic energy.

A property shown by Kelvin to be characteristic of all Dynamical

V] FUNDAMENTALS OF THE IRROTATIONAL FLOW 193

systems started instantaneously from rest is that the kinetic energy
generated is a minimum. The motions calculated in the present
chapter have the least kinetic energy that could arise from the dis-
placement of the body through the fluid.

1 20. Impulse and Kinetic Energy of the Flow Generated by a Normal
Plate

The case of a plate set instantaneously from rest into motion at
right angles to its plane provides an important example of the fore-
going. Assume two-dimensional conditions ; take the origin mid-
way between the edges of the plate, draw Ox in its plane, and let
2c be its width and V its final velocity.

For the impulse I per unit length perpendicular to the #y-plane —

/ = — p J (f> ds . . . (i)

where the integration extends round the whole contour ; i.e. from
one edge along one face round the other edge and back again along
the other face. From Article 117, on the plate, where ^ = 0 —

^ = __ Vc sin Y] . . . (ii)

and Y] ranges from 0 to 2n in the integration. Also from that
article, dx = — c sin YJ dt\. Hence (i) gives —

T2ir

/ = r>Vc* sin1 Y] dr\
i ®

..... (149)

Writing from (ii) sin TJ = — <f>/Vc and from Article 117 x = ~c cos i)
on the plate, where cosh 5 = 1, we have —

showing that the distribution across the plate of <£ and, therefore, of
the impulse is elliptic.

Half the final velocity of the impulse is constant across the plate
and is equal to \V. Hence, from (149) —

E = iTrpF'c* .... (150)

A.D.— 7

Chapter VI
TWO-DIMENSIONAL AEROFOILS

121. The present chapter obtains the streamlines and other
details of irrotational incompressible flow past streamline and aero-
foil sections of practical forms. The process employed is an applica-
tion of the methods of conformal transformation, the aim of which is
to enable the flow in analytically complicated circumstances to be
inferred from that in some simple case whose solution either is known
or can readily be obtained. The method is applicable to two-
dimensional conditions only, so that the shapes derived must be
regarded as the sections of long cylinders whose generating lines are
perpendicular to the #y-plane.

A simple type of conformal transformation will first be described
as an introduction.

Every point in a field of two-dimensional irrotational flow has
attached to it a particular value of z = x + iy and of w = </> -f- ify.
The relationship between these is known at every point, if by the
methods of the preceding chapter we can construct the equation —

»-/(*) : . . (i)

for the flow in question ; ^ and fy are separately obtainable as the
real and imaginary parts, respectively, of the function of z.

We have seen how a particular point z' can immediately be plotted
in the #y-plane, which will now be called the 2-plane. In like manner
w' may be regarded as the complex co-ordinate of the corresponding
point in another plane, called the w-plane, whose rectangular co-
ordinates, instead of being x and yt are $ and i|/. If a region of the
flow in the 2-plane be mapped with a network of equipotentials and
streamlines, the whole network can accordingly be replotted in the
w-plane.

Such an operation is called a transformation. It can only be
carried out by means of a formula connecting the co-ordinates in
the two planes, which is called the transformation formula. The
process is, of course, reversible ; (i) equally enables a network to be
transferred from the w- to the *-plane.

Now, in the simple transformation under discussion the network

194

TWO-DIMENSIONAL AEROFOILS

195

CH. vi]

in the w-plane can have only one form : it must consist of one group
of straight lines parallel to the ^-axis and another group parallel to
the tjj-axis. For equal intervals of <f> and fy the mesh will be square,
whether it be fine or coarse. But this is not true of the corresponding
network obtained by transformation to the z-plane.

Suppose, for instance, that it is desired to transform the square
mesh net in the ie>-plane to the equipotentiafs and streamlines of the
flow at unit velocity without circulation past a circular cylinder of
radius a in the *-plane. We happen to know from Article 113 the
form of (i), which will achieve the result ; it is :

which can be written —

w as z + a*/z .
= re* + «V"*r.

(ii)
(iii)

w- plane

Fig. 82 shows the results
of the transformation
of part of the ze>-plane
above the axis of tf>.
A very small square
element of the te>-plane
will evidently trans-
form to a small square
element of the 2-plane,
although its size,
orientation, and dis-
position geometrically
relative to the axes are
changed. On the other
hand, a larger square
element of the t^-plane
transforms to a dis-
torted figure in the
2-plane. This illus-
trates a characteristic
of conformal trans-
formation : correspond-
ing elements are geo-
metrically similar if
infinitely small, but
not so if finite.

Another point in the present example is as follows
easily obtain, as in Article 113 —

z-plane
FIG. 82.

from (iii) we

1 96 AERODYNAMICS [CH .

<£ = (r -f a*/r) cos 6
fy = (r — a*/r) sin 0
and on the circle, where fy = 0 and r = a,

<f> = 2a cos 6 . . . . (151)

Since 0 varies from 0 to 2n, the maximum and minimum values of <£
on the circle are ± 2a. * Thus the circle itself corresponds to both
sides of a line of length 4a lying on the <£-axis and bisected by the fy-
axis. Moreover, the formula (151) relates each point on this line to
a corresponding point on the circle.

Now, the plot in the z#-plane can be regarded as representing
uniform flow parallel to 0$ past a tangential plate of length 4#.
The formula (ii) then relates at every point this simple flow to the
flow past a circular cylinder of radius a.

Similar results are obtained in dealing with a cylinder of any other
shape if circulation is excluded. But as a rule the form of (ij is not
known. If, however, we can find a means of opening out, as it were,
a part of the <£-axis into some section that interests us, then a proper
generalisation of the process gives the flow past the section. The
example given is fully known in analytical terms. But in other
cases of practical interest analytical treatment might be complicated,
while a solution might more readily be obtainable by graphical
means. An intermediate step is then required, however, as will be
described in the following article.

It may here be remarked, however, that the transformation (i)
becomes of direct use when the real flow in the 2-plane is known, but
is too complicated for the study of some added problem ; the
simplified flow obtained by transformation to the z#-plane may per-
mit of a solution there which can be transformed back. The trans-
formation was so employed by the French engineer Boussinesq in
his pioneering work on heat transfer, and is often known after him.

122. Conformal Transformation

Consider the transformation of part of the z-plane, where the co-
ordinate of a point is z = x + iy, to the corresponding part of a
rf-plane, where the complex co-ordinate of a point is t = £ + nj, so
that the co-ordinates in the J-plane are 5 and yj. Let —

<=/(!) . . . (i)

and assume that throughout the regions considered (i) leads to a
unique relationship between z and t and that dtjdz has a definite
value. Thus for the present we exclude transformation formulae

VI] TWO-DIMENSIONAL AEROFOILS 197

such as t = z*, while also we assume that in the parts of the planes
considered dt/dz has neither zero nor infinite values.

As in Article 112 82, 8z may be interpreted as very small vectors.
Applying the operator dt/dz to an element-vector in the one plane
converts it to an element- vector in the other, and this transformation
is independent of direction. Elementary lengths in the z-plane are

increased on transformation in the ratio

1. Further, element-

dz
lines are rotated through an angle equal to the argument of dt/dz.

It follows at once that angles between adjacent short lines are
unchanged by the transformation, so that infinitesimal correspond-
ing areas are similar. Further, it follows that the magnitudes of

dt »
very small corresponding areas are in the ratio -_ : 1.

ctz

Such a transformation is said to be conformal.
Let <£ and fy, defined by —

w=F(t),

be the velocity potential and stream function of a motion in the
2-plane and let the boundary there be F^Z, 73) = constant. From
(i) we can substitute for t in terms of z and obtain —

w =/(*).

In the same way we can find a new boundary f^x, y) = constant in
the 2-plane corresponding to that in "the £-plane. The same functions
0 and ty then hold for the motion in the #~plane.

Considering a small area mapped with streamlines and equi-
potentials transformed by (i) from the z- to the £-plane, the distances

separating streamlines or equipotentials diminish in the ratio -j : 1.

dl>

Therefore, velocities at corresponding points are increased in that
ratio, i.e. at corresponding points —

To make such increases in local velocity representative of the
change in the boundary shape, we must arrange that the same
velocity exists at infinity in the two planes. If when z is large
dt/dz = 1, the transformation is sometimes known as one-to-one,
but this term often signifies absence of double points.

The distribution of velocity in the 2-plane, say, will be known, and
that in the tf-plane will immediately follow from (162). Application

198 AERODYNAMICS [CH.

of Bernoulli's equation then gives the distribution of pressure in the
/-plane.

123. Singular Points

Reconsidering now the special assumptions made in the last
article, we note first that the transformation formula may be of such
a form that a point in the *-plane, as in the example mentioned,
corresponds to two points in the *-plane, one-half of the one plane
transforming to the whole of the other. The remaining half of the
first plane may then be mapped, if it is required, on a second sheet
of the other. A further example occurs in Article 1 17, if interpreted
in this way, where the whole of the *-plane maps into a strip of the
J-plane of width 2n.

Turning to the second assumption, we shall always find on extend-
ing the region transformed to cover the whole of one of the planes
that certain points occur where dt/dz becomes zero or infinite. Such
points are known as singular points, and the transformation ceases
to be conformal there ; they must either receive special investiga-
tion or be specifically excluded.

An example occurs in Article 121. For clearness rewrite (ii) of
that article as —

t=z + a'/z . . . (i)

so that the ie>-plane of Fig. 82 becomes the tf-plane. Differentiating—

dt/dz = I — «»/**

and there are two singular points where dt/dz = 0, viz. z = ± a ; i.e.
x + iy = ± <* or y = 0, x = ± a. In words, the circle of radius a
cuts the #-axis in two singular points. It is seen at once that the
transformation ceases to be conformal at these points ; for the angle
between adjacent elements of the circle is everywhere TT, while,
although this also holds for the line as a whole, at its ends the angle
becomes 2?u. A singular point is seen to produce a discontinuity in
the transformed contour.

124. Transformation of Circular Cylinder into Normal Plate

An alternative solution of the case of motion investigated in
Article 117 will now be described briefly. The opportunity will be
taken to effect certain calculations required later on, which were left
over in anticipation. The article is of further interest in that in
principle it forms a starting-point for more difficult work than is
attempted in a first reading of the subject.

Flow past a circle of radius a at unit velocity parallel to Oy is

VI] TWO-DIMENSIONAL AEROFOILS 199

obtained from that parallel to Ox by multiplying the co-ordinate by

it giving—

a1 I a*\

- * + 5 — A* ~ T> ' '

The circle itself is transformed into a line of length 4a on the 5-axis
of the /-plane by the formula —

t = z + a*/z . . . (ii)

as we have seen, and transformation of the flow (i) by this formula
will give in the /-plane the flow past a normal plate. To obtain the
potential function of the flow in the /-plane, we require to eliminate
z. Squaring both equations and adding —

w* + t* = 4a*
or —

w = *V/a — 40a . . . (iii)

If u', v' are the £- and vj-components of velocity in the t-plane
from (136)—

dw ,

- = „' - „'.

Hence (iii) gives —

u' — w' =

V{ (£ + »))' -4«'}'
In the plane of the plate 7] = 0 and we have —

' ' for 5 > 2a

(iv)

Hence beyond the edges of the plate, in its plane —

u' = 0, »' = -
while over the surface of the plat

v' = 0, w' = ?/V4af — £' . . . (153)

The sign of «' depends upon which side and which face of the plate
is considered, but is obvious on inspection. The expressions for u'

-0-5 -

200 AERODYNAMICS [CH. VI

and v' have been obtained for unit velocity parallel to the ^/-axis ;
for any velocity V the right-hand sides are to be multiplied by V.
If P is the undisturbed pressure of the stream of velocity F, Ber-
noulli's equation gives for the pressure p over the plate —

P-P __l fi (W8 1

PF' ~*L l-(5/2a)«J-

The further details calculated above for the normal plate will
elucidate remarks made in Articles 117 to 119. For instance, (153)
indicates clearly that the velocity tends to infinity at the edges.
The calculated pressure is as shown in Fig. 83 and, being the same

over both faces, gives zero drag. In
experiment, when a permanent
regime has become established, and
the flow has broken away from the
edges, the whole of the upstream
face has an increased and the down-
stream face a decreased pressure,
leading to the large drag measured.
The result (iii) is not in con-
venient form for plotting, elliptic
co-ordinates being suitable for this
as in Article 117. Instead of
deriving these, an approximate graphical method of general utility
will be described.

Fig. 84 shows in the £-plane the streamlines of (i) and, superposed,
the <£- and i[»-lines of w = z + a2/z- In the /-plane is shown as a
background the entirely square network obtained by transforming
the latter potential function by the formula (ii). Now follow any
streamline of the flow (i). The plotting being close, the streamline
nearly crosses at several points the intersection of a <f>- and a ^-line
of the #-flow. Read off the pairs of values of <f> and fy where this
occurs, and by their use plot points on the square network of the
tf-plane. One of the points so transferred is shown encircled. The
points will be approximately on one of the streamlines of (iii) and a
smooth curve may be drawn through them. The proof is left to the
reader. The graphical method can be used to find the streamlines
of flow without circulation past an inclined flat plate (cf. Article 117).
For this purpose the direction of the flow in the 2-plane to be trans-
formed, instead of being rotated from the #-axis through 90° as in
(i), is set at the appropriate angle.
An alternative graphical method is based on the fact that circles

FIG. 83. — PRESSURE DISTRIBU-
TION OVER A NORMAL PLATE
IN IRROTATIONAL FLOW.

\ \ \

7034-321 01 254-3 8780 10 II ia

5ti

il
O
9
8

-I

-2

-3

-4-

-3

-Q

-7

-IO
-II

t-plane

FIG. 84. — GRAPHICAL METHOD FOR OBTAINING THE STREAMLINES PAST A

NORMAL PLATE.

.D. — 7*

201

202 AERODYNAMICS [CH,

with centres at the origin in the 3-plane and the orthogonal system
of radial lines become ellipses and hyperbolas, respectively, when
transformed to the tf-plane by formula (ii). For substituting
z = aem + tnt which represents circles of radii aem together with
radial lines making angles n with the #-axis, the formula gives —

t =

(m + in) = 2a cosh (m + in).

In the £-plane, therefore, m and n are the elliptic co-ordinates already
employed in Article 117. Thus mapping the 2-plane with a net-
work of such circles and radial lines and the tf-plane with the
corresponding confocal ellipses and hyperbolas provides correspond-
ing systems of co-ordinates which enable any curve drawn in the one
plane to be transformed at once to the other plane.

FIG. 84 A. — ALTERNATIVE GRAPHICAL METHOD.

Fig. 84A illustrates the method in application to the problem
of finding the streamlines of irrotational flow past a plate inclined
at an angle 0. The tf-plane is mapped for equal intervals of m and n,
represented by the proportional numbers 1, 2, 3, . . . , and the
plate is the straight line of length 4a joining the foci. The same
values of m and n yield the network shown in the *-plane, where
both sides of the straight line map into the circle of radius a. The
transformation (ii) is such that the undisturbed streams are inclined
at the same angle 6 to the real axes in both planes. Hence any
streamline may be drawn in the z-plane by Article 108 or otherwise.
Values of m and n for points on this streamline are read off in the
z-plane and replotted in the /-plane, yielding the corresponding
streamline past the inclined plate. The streamlines leading to the
stagnation points are radial for the circle and hyperbolic for the
plate.

VI]

TWO-DIMENSIONAL AEROFOILS

203

SYMMETRICAL STREAMLINE SECTIONS
125. Joukowski Sections
The transformation formula —

t=z + a*/z .... (154)

involves, as has been noted, singular points at x = ± a, marked R
and Q in Fig. 85. To avoid discontinuities in the /-plane contour,
these points must be excluded from the area transformed. This is
achieved by applying
the formula to a
circle of radius > a
enclosing Q and R.
Describing such a
circle with 0 as
centre results in an
ellipse, but displac-
ing its centre a little
upstream leads to a
section of the stream-
line form found in
experiment to give
small drag. The
flow to be trans-
formed will now be
that past this greater FIG. 85.

circle, of centre B

and radius b, which will be called the 6-circle to distinguish it from
the a-circle that yields Q and R.

The form of (154) permits the contour in the /-plane to be found
by a simple construction. Let P, Fig. 85, any point on the 6-circle,
become P' in the /-plane. We have for the co-ordinate of P —

z = re"

and for that of P'—

Thus the co-ordinate of P' is the sum of two vectors, the first of
which is identical with the vector OP. Dealing with the second
component vector, the modulus a*/r means * that P is to be reflected
in the a-circle, giving Plt while the argument — 0 means that OPl

* The relation OP . OPl «• a* is clearly necessary. Cf . also Art. 162.

204 AERODYNAMICS [CH.

so obtained is to be reflected in the #-axis, giving OP2. The vector
OP' is found by completing the parallelogram POPZP'.

This graphical method can be applied, of course, to points outside
the 6-circle, so that any point on any streamline past the circle can
immediately be transformed to the /-plane in the same way, its
radius / being written for r.

FIG. 86. — THIN JOUKOWSKI SECTION, BOTH POLES EXCLUDED.

One-half of the section is magnified transversely in the lower diagram to show

details.

In Fig. 86 b/a = 1-05, OB/a =0-035. The transformed section
is of thin symmetrical streamline form, such as might be adopted for
an aeroplane fin or tail-plane. Half the contour is also plotted with
its thickness magnified ten times to show the slight rounding
achieved at the trailing edge which is necessary for practical con-
struction. Another point of practical interest is that an appreciable
length of the rear part of the contour is very nearly straight.

Fig. 87 shows the streamlines round a thick section suitable for a
strut, drawn by the same method. Here b/a = 1-24, OB/a = 0-1 86.

FIG. 87.— STREAMLINES PAST A JOUKOWSKI STRUT.

If the i-circle be so drawn as to enclose R only, passing through Q,
a sharp trailing edge is obtained, as illustrated in Fig. 88. The

FIG. 88. — STANDARD JOUKOWSKI SYMMETRICAL
SECTION, ONE POLE ONLY EXCLUDED.

VI] TWO-DIMENSIONAL AEROFOILS 205

trailing edge is infinitely

thin, both surfaces having

a common tangent there,

while the rear parts of

the contour are concave

outwards, making an

unpractical shape. On the other hand, the section is analytically

simple, and in some calculations may be substituted for a more

complicated shape without serious error. A theoretical interest

will also appear later in the sharp trailing edge. Unless otherwise

stated, it is this particular type of section which will be referred

to as a Joukowski symmetrical aerofoil.

126. Approximate Dimensions

In many Aerodynamic applications of the foregoing the lines of
sections are ' fine ' (cf. Article 107) and the reciprocal of the fineness

ratio, called the thick-
ness ratio, is then intro-
duced. Thickness ratio is
accordingly defined as the
ratio of the maximum
thickness of the section
to the chord. For small
thickness ratios b is little
greater than a, and
certain dimensions may
be evaluated for the
Joukowski symmetrical
aerofoil.
Let--

b=a(l+m) . (i)
where m is small compared

FIG. 89.

with a. Considering the point P (r, 6), Fig. 89-

(r sin 0)a + (r cos 6 — am)* = b* = a* (I + m)*.
Neglecting the terms (am)* because m is small, this gives —

r* — Zram cos 0 = a* (1 + 2m) ;
or —

— 2m cos 6 . (1 + 2m)

206 AERODYNAMICS [CH.

Now r/a is positive. Again neglecting terms of order m* —

- = m cos 6 + \/(\ + 2m)
a

and— a

= 1 + m (1 + cos 6)

= i _. m (i + cos 6) . . (iii)

r
approximately.

Hence, in the *-plane we have for P', the point corresponding to P,
remembering (154) —

E= a cos 0 ( r + - ) = 2a cos 8
\a r/

/r a\

Y) = a sin 0 I 1 = 2am sin 0(1+ cos 0)

\a r/

(iv)

The first of these formulae states that when m is small compared
with at the chord of the section is 4a to the first order. Again, the
thickness ratio is the maximum value of 2-y]/4a, and differentiating
the right-hand side of the second formula with respect to 0 and
equating to zero gives cos 0 = £, so that sin 0 = £\/3. Hence :

Thickness ratio = m . - (1 + £)

= —^— m = 1-3 m, approximately. (165)

The maximum thickness, occurring when cos 0 = £, i.e. when £ = a,
is situated at one-quarter of the chord from the leading edge.

Eliminating 0 leads to a simple formula by which narrow aerofoils
of the cusped form shown in Fig. 88 can be plotted directly.

Let X be the distance from the trailing edge of a point on the
chord-line, expressed non-dimensionally in terms of the chord.
Then the first of (iv) gives —

x = 21-U = K! + cos m.

Hence if Y denotes the ordinate at X, similarly expressed, the
second of (iv) gives —

y = 1 = mx sin 0
4a

= 2mX*!*(l - X)1/3 . . . (156)

Rounded-tail Aerofoils. — Narrow sections derived by the Joukowski
transformation from an eccentric circle enclosing both singular
points, as in Fig. 85, can be treated similarly.

VI] TWO-DIMENSIONAL AEROFOILS 207

Let b = a(l + m) as before, and let OB = a/, where'/ < m.
The following expressions result in place of (iii) —

f m + / cos 0

~ = I — m — I cos 6.

Thus the first of (iv) remains unchanged, but the second becomes —

Y) = 2a sin Q(m + / cos 6). . . (v)
Introducing X and Y as defined above and substituting,
y = £{i _ (2X ~ !)•}«»{ w + l(2X - 1)}

- (2/X + m - /) X1/3 (1 - X)1/3 . (vi)
The last expression can be rearranged as —

Y = 2/X3/2 (1 - X)1/3 + (m - QX1^! - X)1'3. (157)
The first term on the right has the same form as in (156) and thus
represents a thinner-cusped aerofoil. The second term is an
ellipse. Hence the rounded-tail Joukowski symmetrical section can
be described as a cusped aerofoil of reduced thickness enveloping,
or built round, a core consisting of an ellipse of the same chord.
The position of maximum thickness no longer occurs at one-quarter
of the chord from the nose but farther back, depending upon the
ratio m/l. Let X' denote the value of X for maximum thickness.
Then by differentiating (157) and 'equating to zero,
1 _ gx'(i - X') f»

i~2X' -7' * ' (vn)

For mil = 1, X1 = f, as already found. Reducing X' from this
value soon causes mil to increase rapidly, leading to predominance
of the elliptic term in (157) and consequently to a notably blunt tail.
The curve (a) of Fig. 90 is the half-profile of a symmetrical Joukowski

JTICK 9o. THICKNESS DISTRIBUTION OF SYMMETRICAL AEROFOILS WITH MAXIMUM

THICKNESS AT 0'4 CHORD FROM NOSE.

(a) Joukowski, (b) K&rmdn-Trefftz, (c) Piercy.

aerofoil having a thickness ratio of (H5 and the position of maximum
thickness located at 0-4 of the chord from the nose (Xf = 0-6) ; in
this case m\l = 4-6. The curves (b) and (c) will be described later.

208

AERODYNAMICS

[CH.

It will, of course, be noted that m is no longer connected with the
thickness ratio by (155).

1 26 A. Velocity and Pressure

The velocity qt at any point in the flow past a symmetrical
Joukowski section at zero incidence is calculated as follows :

The first step is to determine the point P (r, 0) in the z-plane of
the circle corresponding to the given point £, T) in the 2-plane of the
aerofoil. The transformation formula —

i = I + * V) = z + a*/z . . (i)

gives on separation of real and imaginary parts —

5 = (r + a*/r) cos 6, TJ = (r — a*/r) sin 6 . (ii)
and combining these leads to —

cos 0

sin*0

+ -r^ = 2r.
sin 0

(iii)

0 is found from the first of (iii) and then r from the second.

The undisturbed velocity in the 2-plane is taken as — U and the
circle as of radius b. The potential function of the flow past the
circle is then —

w - <£ + ty = -U(zl + 6V*i). • (iv)

FIG. 90A.

in which zl = xl + iyl = ^d1^ is the complex co-ordinate of a
point referred to axes with B as origin parallel to Ox, Oy, B being the

VI] TWO-DIMENSIONAL AEROFOILS 209

centre of the circle, Fig. 90A. Considering the "projections of OP,
BP in the figure,

so that —

tan 0X =

r cos 0 = r
r sin 0 = r

r sin 0

cos
sin

r sin 0
sin "07

(v)

r cos 0 - OB' ]

These together with (ii) enable the co-ordinate zl corresponding to
the point given in the aerofoil-plane to be found.

The velocity qn at P can now be obtained from (iv) by (iii) of
Article 113A—

- U(l -

b* . A

4- 4 — sm« 6,

(vi)

The transformation gives —

dt/dz = 1 — a*/z* = 1 when z is large. . (vii)

Hence the undisturbed velocity is — U in the ^-plane as well as in
the 2-plane, and the velocity at the general point is given by (152)
of Article 122, i.e. —

ft =

In the same manner as for (vi) it is found that —

dt
dz

. (viii)

the similarity to (vi), a feature of the Joukowski transformation,
being due to the similarity between (i) and (iv).

These formulae are general. But an important special case arises
when the given point is on the profile of an aerofoil of normal
thickness with a cusped tail and the corresponding point on the
circular boundary. Then (vi) reduces to qu = 2 17 sin Qv and 1 —
2m(l + cos 0) can be substituted for a*/r*.

Finally, if P is the undisturbed pressure and pt the pressure at
the given point, Bernoulli's equation gives —

(ix)

210 AERODYNAMICS [CH.

127. K4rm4n-Trefftz Symmetrical Sections

It has been seen that Joukowski sections suffer from practical
limitations, briefly as follows. If the tail is sharp it is also cusped,
and the shape of the profile is controlled by only one parameter,
viz. m, which varies the thickness ratio ; the position along the
chord at which the maximum thickness of the section occurs is
invariable and too far forward. Admitting a rounded tail is in-
effectual because the tail becomes blunt, causing much form drag,
when the position of maximum thickness is moved back appreciably.
To overcome these and other drawbacks calls for more elaborate
transformations .

An early improvement provided profiles which are known as
extended or generalised Joukowski aerofoils, or after K4rm&n and
Trefftz. The formula (154) is identical with—

_ (z +
~V7

which is a special case of the transformation —

t — na z — a

whose singular points are at z = ±_ a as before. Using (158) to
transform a &-circle drawn through one of the singular points and
enclosing the other, as shown in Fig. 89, enables the aerofoil to be
given a ' tail angle ' T, defined as the angle at which the two sides
of the section meet at the tail. The angle is secured by choosing
for n a value less than 2 according to the relation —

n = 2 — T/TT. . . . (ii)

Again, the position of maximum thickness can be adjusted while

FIG. 01. — KARMAN-TREFFTZ SECTIONS,

retaining a tail angle by suitably relating b[a to n ; for example,
2n = 5 — bja locates this position at one-third of the chord from

VI] TWO-DIMENSIONAL AEROFOILS 211

the nose. Two sections shown in Fig. 91 have the following
characteristics :

Section

»/•

Thickness

A ...

1-967

1-067

10 per cent.

1-833

1-333

40 per cent.

The first might be suitable for a tail-plane ; the second, better
described as having a fineness ratio of 2-5, is rather thicker than
would be used for a strut. But modern conditions usually require
the maximum thickness to be located still farther back. The
transformation (158) is insufficiently elastic from this point of view,
as will be illustrated. Moreover, the simplicity distinguishing the
Joukowski transformation is lost ; (158) is best dealt with, indeed,
as a special case of a more general transformation whose discussion
is beyond the scope of this book. In these circumstances, the
detailed treatment of these sections is left to further reading.*

It can be shown, however, that narrow Kdrmdn-Trefftz sections
accord closely with the formula —

y =

+ sx(i

(159)

where X and Y have the meanings defined in Article 126 and c and
s are two parameters. The first term on the right is seen to have
the same form as in (156), and the second term is a circular arc.
Thus the section can be described as a cusped Joukowski aerofoil
built round a core of the same chord formed by two segments of a
circle. The non-dimensional distance X' of the position of maximum
thickness from the tail is given by —

As the position of maximum thickness is moved backward, c/s
decreases, showing, in conjunction with (169), that the circular arc
then tends to control the shape except close to the nose. The
result is a flattening of the front part of the profile as illustrated
by the half-profile (b) of Fig. 90, for which the maximum thickness
is located at 0-4 of the chord from the nose (c/s = 0-544).

* Glauert, A.R.C.R. & M., No. 911 ; Page, Falkner and Walker, A.R.C.R. & M.t
No. 1241.

212

AERODYNAMICS

[CH.

128. Aerofoils inverted from Hyperbolas

A more amenable family * of aerofoils avoiding the defects of the
Joukowski system is obtained by inverting one branch of an
hyperbola. The shape is controlled by two independent parameters
which may be arranged to secure a prescribed tail angle and position
of maximum thickness of the section. The latter can usefully be
varied between 0-3 and 0-45 of the chord from the nose (for farther
back positions the nose sharpens rapidly). A description of the
symmetrical form of this family is given in the following articles
and provides an introduction to 'methods used in more advanced
work, where the number of parameters is further increased.

FIG. 92. — THEZO-PLANE.

Fig. 92 shows the two branches of an hyperbola whose centre is
at the origin of a z0-plane and whose ' transverse axis ' AB lies on
the #0-axis. The foci are jFA, FB and the angle between the asymp-
totes is T. It is one of the family represented by the equation
^/cos2 ^ - yll&rffa = 1 ; OFA = 1 and therefore OA = cos <r/2.
The right-hand branch will be transformed into an aerofoil, for which
purpose the complex co-ordinate 20 == XQ + iy0 is suitable, but it will
also be transformed into a circle and for this operation the complex
co-ordinate is changed to £ = pi + iv by the formula z0 = cosh £.
Relations are readily found as in Articles 115 and 117 between x0t
y0 and p, v, but the new co-ordinates differ from those of Article 117
in that (j. may assume any real value, positive or negative, whilst v
is restricted to lie within the range O to TU. It follows that v = con-
stant gives one of a system of confocal hyperbolas, the constant
being equal to one-half the angle between the asymptotes, and that
y. = constant gives the upper or lower half of one of a system of

* Piercy, Piper and Preston, Phil. Mag., Ser. 7, vol. xxiv, p. 425 (1937). Piercy,
Piper and Whitehead, Aircraft Engineering, November, 1938 ; Piper, Phil. Mag.,
Ser. 7, vol. xxiv, p. 1114 (1937). For further generalisation and applications see
later publications by Piercy and Whitehead (when released).

VI]

TWO-DIMENSIONAL AEROFOILS

213

confocal ellipses, according to whether the latter constant is positive
or negative, respectively.

Some other values of the co-ordinates are indicated in the figure.
Along the #o-axis : v = 0 and the sign of IJL is indeterminate from
FA to infinity, v increases from 0 to K and \L == 0 from jpA to FB,
v = 7i and the sign of p is indeterminate from jFB to — oo. Along
fyo> v = */2 and P1 increases from 0 to oo ; along — Oy0, v = Tc/2
and (A decreases from 0 to — oo.

The hyperbola v = T/2 will be inverted with respect to a centre
of inversion C located on the #0-axis at a suitable distance e from 0,
reckoned positive if C is to
the left of the origin in the
figure. Co-ordinates of C
are distinguished by suffix c.
If e < 1, C lies between ir/2
and in since pc = 0. If
e > 1, £. = [ic + in, giving
z0c = — cosh (JLC, and this
quantity is determinate al-
though (ji itself is rendered
uncertain by the change of
sign on crossing the #0-axis
beyond the focus.

The right-hand branch of
the hyperbola is replotted
in the ^-plane of Fig. 92A,
where the origin Ol is coinci-

t -plane

z, -pi tine

2 -plane
FIG. 92A.—

z2 -plane

TO CIRCLE

dent with the centre

inversion C. Thus with

= 1, the complex co-ordinate in this plane would be z± = %l + iy-i

= s + cosh ^, but a change of scale is made below to O^A = 1, as

marked in the figure.

In the £-plane of Fig. 92A is shown a symmetrical aerofoil obtained
from the hyperbola by the formula —

tzl = 1. . . . (160)

Substituting t = rteiet, zl = r^i leads at once to—

rt - l/rlt 6, - -8V . . . (i)

Thus points remote from the origin in the ^-plane are close to the
origin in the tf-plane, and vice versa ; remote parts of the hyperbola
yield the back part of the aerofoil, and the part of the hyperbola in
the neighbourhood of its vertex provides the rounded nose of the

214 AERODYNAMICS [CH.

aerofoil. The inversion is accompanied by reflection in the real
axis, so that the upper side of the aerofoil corresponds to the lower
side of the hyperbola, and vice versa. Thus jx is negative on the
upper side of the aerofoil profile.

The chord c of the aerofoil is equal to the inverse of the #raxis
beyond A, i.e. 1/0^4, and this is made equal to unity by multiplying
lengths in the ^-plane by l/(e + cos T/2). Then the distance of C
to the left of the centre of the hyperbola becomes e/(s + cos T/2),
and the complex co-ordinate of the general point in the zrplane
becomes —

^e±^ . . . .(161)

1 e + cos T/2 v '

It will be seen that the tail angle T of the aerofoil is equal to the
angle T between the asymptotes of the hyperbola. T and e comprise
the two independent parameters of the family.

Plotting the Aerofoil Profile. — Any point on the hyperbolic
boundary will be denoted by Xv Yl and any point on the aerofoil
profile by X, Y ; the co-ordinates are non-dimensionally expressed
in terms of the chord, and X, Y have the same meanings as in
Articles 126 and 127. With this notation, (160) gives—

X + iY = l/(Xl + * yx) . . (ii)

i.e., on rationalising the denominator on the right by multiplying
by Xl — iYl and separating real and imaginary parts,

X = XJRf and Y = -YJRf . (iii)

where R^ = X^ + Y^, and to map the aerofoil we have only to
determine Xl and Yr (161) gives, on separating real and imaginary
parts,

__ s + cosh LL cos T/2 _ „ sinh LL sin T/2 .. .

Xl = — . and YI = — £ ±- (iv)

1 e + cos T/2 * e + cos T/2 v '

for a chosen position of the centre of inversion and tail angle T of
the aerofoil. Eliminating \i yields the following relations between
Xi and Y! :

yt- = x-^ - i)^ + &) . (v)

where,

x == tan T/2 and b = — ^—^ C° T{ . (vi)
1 e + cos T/2 v ;

The above method is exact, but an approximate formula which is
more direct can usually be employed instead.

Vl] TWO-DIMENSIONAL AEROFOILS 215

Noting that if R* = X> + Y» (ii) equally gives Xl = X//?», etc.,
and substituting,

*-«•(*-)(*+')• • <->

This expression also is exact. But it contains terms in Y4 which
may usually be neglected, leading to the approximate formula —

(188)

bX)}' V ;

Further approximation is permissible in the case of very thin aero-
foils, for which the denominator will differ little from unity, so
that—

y = ± xX(l - X)W(I + bX)112. . (viii)

But (162) should be employed for aerofoils having thickness ratios
within the range 0-12 to 0-20 common in practice.

Parameters and Shape. — It is usually required to determine the
parameters x and ft, whence T and e follow, for an aerofoil of chosen
thickness ratio and position of maximum thickness. The condition
for a maximum ordinate, i.e. dY/dX — 0, may be expressed as- —
1 dY± _ 2Xl
Y^'dX,- Xf^Tj

A second expression is obtained from (v), for differentiating both
sides of that equation with respect to X^ and dividing by ZYf
gives —

* (X.-l^X. + b) •
Equating the two expressions leads to —

_ ^'(2^ - 1) + X1'(2X1 - 3)

X&=3XJ=Y{

where the point Xv Yl corresponds to the chosen co-ordinates X',
Y' of the aerofoil profile at its position of maximum thickness.
For an aerofoil so thin that Y^ may be neglected, giving X =
l/Xl approximately, (ix) reduces further to —

2 - 3X'

The curve (c) of Fig. 90 is the half-profile of an aerofoil of the
present family with the position of maximum thickness located at
0-4 of the chord from the nose, and may be compared with the
corresponding profiles (a) and (b) for the Joukowski and K£rm4n-
Trefftz families, respectively, which have already been described.

216 AERODYNAMICS [CH.

129. Completion of the Transformation

The aerofoil cannot be transformed into a circle directly but only
through the hyperbola, which is changed first into an infinite
straight line in a z2-plane, and then into the circle in a 2-plane,
Fig. 92A.

The first step is accomplished by the formula —

. . (163)

where % has already been defined, being the complex p + iv in which
v is restricted to lie between O and TC. e" is a constant such as to
ensure that the origins in the 2r and z2-planes shall be corresponding
points. Hence, putting z^ = 0 when z2 = 0,

- • • »

This transformation may be regarded as changing the given hyper-
bola into the hyperbola which coincides with the jy0-axis in the
20-plane, Fig. 92. However, in the 22-plane it is defined by v = r/2,
and the formula (163) arranges that the origin in this plane is at
unit distance to the left of the straight line, as marked in Fig. 92A.
A circle inverts into a straight line if the centre of inversion lies
upon the circle, and the formula —

z2(z + 1) - 2 . . . (164)

inverts a circle of unit radius with centre at the origin in the z-plane
into the straight line of the z2-plane, and then the centre of inversion
is at the point on the circle which corresponds to the origin in the
tf-plane.

This completes the transformation of the aerofoil of unit chord
into the circle of unit radius. In the reverse order, (164) opens out
the circle into an infinite straight line, (163) and (161) turn the
straight line into one branch of an hyperbola, and (160) inverts the
hyperbola into the aerofoil.

To enable the flow past the aerofoil to be inferred from the
simple flow past the circle, the above process must conformally
transform the region exterior to the circle into the region exterior
to the aerofoil ; all singularities must be excluded from these two
regions except only the singularity yielding the sharp tail of the
aerofoil. (164) transforms the region exterior to the circle into
the region to the left of the infinite straight line in the *2-plane,
giving a singularity at the origin in the z2-plane. (163) and (161)

VI]

TWO-DIMENSIONAL AEROFOILS

217

transform this region into the entire region to the left of (or outside)
the right-hand branch of the hyperbola, introducing no further
singularity in the region considered not on the boundary. The
singularity at the origin in the zr and z2-planes occurs at correspond-
ing points in those two planes and at infinity in the z- and 2-planes,
while the singularity on the circle and aerofoil boundaries occurs at
corresponding points in the z- and ^-planes and at infinity in the
others.

The foregoing may "be illustrated by considering the nature of
the flow in each plane. The uniform flow at a large distance
from the boundary in the z-plane becomes on inversion a doublet at
the origin in the z2-plane. The transformation from the 22-plane
to the ^-plane carries over this doublet to the origin in the 2rplane,
only its strength being changed. The final inversion into the
aerofoil plane reconverts the doublet into a uniform flow at infinity
in that plane, though not of the same velocity as the uniform flow
in the 2-plane. The change of velocity between the circle and
aerofoil planes must be allowed for but is easily determined, as in
the next article.

I2QA. Velocity on the Aerofoil Boundary

Calculation of the velocity in the £-plane from that in the z-plane
requires in the first place a relationship between the positions of
corresponding points. For any point on the aerofoil boundary,
pi can be found from (iii) and (iv) of Article 128, and this boundary
value of \i is related as follows to the corresponding angle 6 in the
circle plane. Substituting z = eie (since r = 1 on the circle) in
(164) and, in the same equation, expressing z2 in terms of £ = (x +
n/2 from (163),

. e i . (ji

tan - - = smh .

2 t" 2 — T/7C

The next step is to evaluate mod. dz/dt from —

fa
dz9

The transformation formulae give —

E - - «* + '>'

cosh

dz,

- ft/2

(i)

(ii)

C'(2 - T/7C) 2 - T/7T

218

AERODYNAMICS

cos T/2

sinh £

[CH.

whence (ii) yields after reduction —

- A cos' cosh

*l'

where the constant coefficient has the value —

_ 2(e + cos T/2)

4& —~

(iii)

. (iv)

e*(2 - T/TT) '

The velocity — U at infinity in the aerofoil plane is derived from
the velocity — U' at infinity in the circle plane by —

-U = -U'

(v)

For z and t large, zl and z% are the co-ordinates of the centre of
inversion, and (ii) gives —

cosh

2 -T/7C

(vi)

sinh

But from (v) we can write, owing to the large values of t and z,

+1
and substituting in (vi) gives —

[-1 —-

L<&J oo z z

4
A

\ sinh

- »T/2

n /r/TC

which reduces to —
dz

- I

. (vii)

Thus the proportionate increase of velocity from the undisturbed
speed U in the aerofoil plane, at the point t corresponding to the
point z in the circle-plane where the ratio qxf U' is known, is given by —

(165)

where the modulus on the right is given by (iii) and A by (iv).

U' '

:"• - l\1/a

~ U' '

• \*-lJ '

4' '

VI]

TWO-DIMENSIONAL AEROFOILS

219

FIG.

92s. — PRESSURE DISTRIBUTIONS
OF THEORY AND EXPERIMENT

COMPARED FOR THE SECTION B

OF FIG. 91.

1296. Comparison with Experiment and Example

Question arises as to how far calculations of velocity based upon
the assumption of wholly irrotational flow agree with experiment
in the case of streamline sections. A comparison between theory
and experiment has been made*
at the National Physical Labora-
tory in the case of the very thick
Kdrm&n-Trefftz aerofoil B of
Fig. 91. The theoretical pres-
sure distribution, ignoring the
boundary layer and wake, is
shown as the full-line in Fig.
92B . Experimental observations
for this section obtained at a
Reynolds number of 6 x 105
gave the broken line. Agree-
ment is seen to be close over
80 per cent, of the contour. With a small thickness ratio experi-
ment still diverges from the present theory as the tailing edge is

approached, but to a much less
extent than in the extreme case
illustrated. Equally successful
comparisons have also been
made with symmetrical sections
of the simple Joukowski type.
The important conclusion is
that for the Reynolds numbers
of Aeronautics the present
methods enable reliable calcu-
lations to be made, except near
the trailing edge, of the pres-
sure round derived shapes of
streamline form, and of the
velocity field outside their
boundary layers. f

In these circumstances the
theory finds many Aerody-

* Loc. cit.t page 211.

f According to Piercy, Preston and Whitehead, Phil. Mag.t Ser. 7, vol. xxvi,
p. 802 (1938), approximate allowance can be made for the wake of a bluff section
by determining the potential function as for an imaginary elongated boundary, in
which the back of the section is replaced by a narrow extension to infinity, represent-
ing the wake.

FIG. 92c. — PRESSURE DISTRIBUTIONS FOR

MAXIMUM THICKNESS LOCATED AT

(a) 0-35 CHORD AND (b) 0-425 CHORD

FROM NOSE.

220 AERODYNAMICS [CH.

namical applications, one of which is indicated in Fig. 92c. In the
figure there have been drawn two examples of the family of aerofoils
inverted from hyperbolas. Both have a thickness ratio of 0-15,
but for (a) the position of maximum thickness is at 0-35 chord from
the nose, while for (b) it is at 0*425 chord from the nose. The
distribution of the theoretical pressure distribution round the two
boundaries is also shown and can be relied upon to agree fairly with
experiment except in the region of the tail. The difference illustrates
a decrease in the maximum velocity ratio achieved by displacing the
position of maximum thickness backward. This decrease and the
backward displacement of the position round the profile at which
the maximum velocity occurs are of importance in designing
sections for low drag and high speeds.

DERIVED WING SECTIONS
130. Circular Arc Skeletons — Joukowski Transformations

The straight lines to which the circle of radius a transforms by
formulae (154) or (158) are known as the skeletons of the symmetrical
sections given by these formulae when applied to a circle of greater
radius b with centre on the #-axis. Skeletons of arched form are
obtained by locating the centre B of the 6-circle on the 3/-axis and
drawing the 6-circle through both the singular points Q and R,
x = -j- a,

Dealing first with formula (154), and applying it to any point P
(rt 0) on the 6-circle so drawn (Fig. 93), we have as before for the co-

FIG. 93.

ordinates of the corresponding point Pr in the £-plane —

£ = (r + a*/r) cos 0 .

TJ = (r — a*/r) sin 0. . .

(i)
(ii)

VI] TWO-DIMENSIONAL AEROFOILS 221

From the triangles OQB, OBP with (i as shown—

6a = aa see1 p = ?* + a* tan* p — 2ra tan p sin 6
or —

r — a*/r = 2a tan p sin 8
Hence from (ii) :

T) = 2a tan p sin2 0 . „ (iii)

showing that two points on the 6-circle at ± 6 transform to a single
point in the £-plane, and that the maximum ordinate of the trans-
formed curve is situated on the yj-axis (6 = ju/2) and equals 2 . OB.
Q and R transform to Qf and R' (Fig. 93), giving Q'R' = 40, as
seen from (i) and (ii). The ratio of the maximum ordinate of the
arch to its chord is called the camber and from (iii) equals £ tan p.
Squaring (i) and (ii) and by subtraction we find—

-_ --- _ -_-

cos8 0 sin2 0
and eliminating 0 by (iii) gives for the equation of the arch—

£2 + (T) + 2a cot 2(3)* == (2a cosec 2p)» . (iv)
a circle whose centre is on the vj-axis at 73 = — 2a cot 2p. The
tangent at Q' is inclined at the angle 2p to the £-axis.

Whilst the formula (154) thus transforms the 6-circle passing
through Q and R to both sides of a circular arc, formula (158) trans-
forms it into two circular arcs (Fig. 94), which intersect at Q'9 R'

at the tail angle T = TT (2 — n). The figure is readily obtained by
the methods of Article 127.*

These and other arched skeletons may be used to bend symmetrical
aerofoil sections into cambered wing shapes. The modulus is not
then known, however,

131. Joukowski Wing Sections

We now consider in some detail wing sections of a certain type intro-
duced by Joukowski in 1910, which are susceptible to simple analysis.

To obtain these the formula (154) is applied to a 6-circle passing

* The transformation is known after Kutta. Detailed investigation of this and
other shapes is given in a paper by Mrs. Glauert, Jour. R.Ae.S., July 1923, which
should be read.

222

AERODYNAMICS

[CH.

through one of the singular points, Q say, and enclosing the other,
with centre B slightly displaced from both axes. For a section of
normal proportions to result, the angle (3 which QB makes with Ox
requires to be small and EB (Fig. 96) a small fraction of a.

FIG. 95. — CONSTRUCTION FOR JOUKOWSKI CAMBERED WING.

A transformed profile of this class is shown in the figure. A
point P' on it is found from the corresponding point P (r, 0) on the
6-circle in the 2-plane exactly as described in Article 125. It may
be noted that the locus of Pt9 the reflexion of P in the a-circle as it
moves round the 6-circle, is another circle of radius < a whose centre
lies on BO produced. The image in the #-axis of the centre of this
latter circle, the point A on QB, is the centre of the equal circular
locus of P2, the reflexion of PI in the #-axis. The circle with
centre A is called the auxiliary circle, and has a common tangent
with the 6-circle at Q. It is easily found that OA, OB make equal
angles with Oy. With the help of the auxiliary circle, the locus of
P', i.e. the aerofoil contour in the /-plane, is plotted rapidly.

Joukowski wing sections are infinitely thin at the trailing edge,
like the corresponding symmetrical sections, as is evident from the
preceding article.

132. Approximate formulae for the co-ordinates 5, >) of any point
P' on the wing are found as follows :

Let m be the small fraction that EB is of a, so that, since (3 is small
we have approximately —

6«fl(l+m). . . . (i)

VI]

TWO-DIMENSIONAL AEROFOILS

223

FIG. 96.

From Fig. 96 and (i) —

(PN)* = (r sin 8 — ma sin (3 — a tan p)a = ra sin1 8 — 2ra$ sin 8
(J3iV)a = (r cos 8 — ma cos (i)a = ra cos2 6 — 2ram cos 8

(BP)* = 6a = ((XB)a = (a sec p + ma)a = a1 + 2waa.

The right-hand members are obtained by taking account of m and (3

being small and neglecting terms of smaller order.
Hence, since (PN)* + (BN)* = (BP)*,

ra — 2ra (p sin 8 + m cos 6) = a* (I + 2m)

or —

/r\a r

( - ) — 2 - ((J sin 8 + m cos 8) — 1 — 2m = 0.

This gives :

- = 1 + p sin 6 + m (1 + cos 6)

- = 1 — p sin 6 — m (1 -f cos 8)

to the first order.
Finally—

= a{- + ~)
\a r/

/> «\ .
= a ( --- 1 si

\a r/

cos 6 = 2a cos 6

(ii)

. .
sin 8

+ cos 8)

in 8}.

(166)

224 AERODYNAMICS [CH.

These formulae may be compared with those of Article 126 for a
symmetrical Joukowski section. The first shows that the £-ordinate
is the same to the approximation considered and the chord equal to
4a as before. The Yj-ordinate is increased by the term 2#p sin* 6.

133. Shape of Joukowski Wings

The shape depends upon the particular values assumed for m and
P. Provided always that these are small, certain characteristics may
be conveniently expressed.

It is first seen that the thickness ratio is still given by (155). For,
if v], T)' are the ordinates of points on the upper and lower surfaces at
any distance along the chord specified by £, the thickness T at that
position is given by —

r = ,,-v

and if TJ is transformed from a point on the ft-circle whose radius
makes an angle 0 with 0%, the corresponding angle leading to 7)'
will be — 0. Hence from (166) of the preceding article :

T = 4am sin 0 (1 + cos 0)

and, on comparison with Article 126 (iv), the result follows. The
maximum thickness again occurs at one-quarter of the chord from
the leading edge.

The mean camber is defined by the maximum value of £ (7) -f- 73')
divided by the chord, or (YJ + V)/8# for m, (1 small, and from the
preceding article :

T) + 7)' = 40p sin2 0.

The maximum value of this, occurring when 0 = |TT, is 4a(3. Hence :

Mean camber = |(3 . . . (167)
as is seen alternatively from Article 130.

Karman-Trefftz Aerofoils

Wing sections of the generalised Joukowski type with finite tail
angle result from transforming a 6-circle whose centre is offset from
both axes in the z-plane by the formula (158). The process
is facilitated by the formulae developed in the papers to which
reference has already been made. K&rmdn-Trefftz cambered aero-
foils have recently been developed further by introducing an addi-
tional parameter.*

* Betz and Keune, Jahrbuch d. LFF., 1937.

VI]

TWO-DIMENSIONAL AEROFOILS

225

1 33 A. Cambered Aerofoils Inverted from Hyperbolas

Cambered aerofoils closely resembling the sections of modern
wings result from inverting hyperbolas with respect to a centre of
inversion which is displaced from the axis of symmetry. The
transformation into a circle requires only slight modification of the
formulae given in Articles 128-9 for symmetrical aerofoils of the
family.

FIG. 96A.

Referring to the ^-plane of Fig. 92A, let the axis of symmetry of
the right-hand branch of the chosen hyperbola be displaced parallel
to itself through a distance Ayx =* 8 such that the origin in this
plane may still coincide with the centre of inversion. Fig. 96A
illustrates the modification. The complex zl is then related to £ by —

e -MS + coshi;

1 £ f COS T/2 V '

in place of (161).

An aerofoil of zero thickness is obtained in the ^-plane when the
hyperbola degenerates into both sides of the part of the axis of
symmetry beyond the focus, i.e. the line v = 0 (cf. Article 128).
This straight line inverts into a circular arc in the ^-plane.

A cambered aerofoil of small thickness results from v = T/2
where T is small, and the above circular arc approximates closely
to the median line of its section and is therefore called its camber-
line. The camber-line may be slightly extended to intersect the
aerofoil profile at the nose, the extension representing the inversion
of the short length FhA of the #raxis, and then the co-ordinate of
the front end of the camber-line, called the nose of the aerofoil, is
£ = fT/2. The part of the line OtA beyond the vertex A of the

A.D. — 8

226 AERODYNAMICS [CH.

hyperbola inverts into the chord-line of the cambered aerofoil, and
the angle 2(3 between this line and FAA remains unchanged by the
transformation close to the nose of the aerofoil. Hence, at the nose
the camber-line makes with the chord-line the angle 2(3 defined by
(see Fig. 96A) —

— ~

e + cos T/2

and it follows that the amount of camber, or the mean camber, is

£p as in (167).

For constant values of the parameters e and T, the maximum

thickness and its position along the chord of an aerofoil are only

slightly affected by camber.
Thus appropriate values for
the parameters may be deter-
mined by means of the formulae
already given for symmetrical
sections of the family , whence
8 follows on choosing the
camber. If a bi-convex section
is desired, the camber must be
so restricted that the centre
of inversion lies between the
asymptotes produced of the
hyperbola, i.e. 8 must be less

(b)

f FIG. 96B. — EXAMPLES OF CAMBERED
SECTIONS.

The value of 5 is greater for (a) than for (b).

than e tan T/2.

Retaining the same change
of scale between the zQ- and
zrplanes, as adopted for the
symmetrical sections , results in
the distance between the

centre of inversion and the vertex of the hyperbola being no
longer equal to unity. The inversion formula (160) is accordingly
modified for cambered aerofoils to —

tzl = 1 + i tan 2(3 . . . (160A)

in order that they shall have unit chord. The change also rotates
the aerofoil through the angle 2(3 so that the real axis of the tf-plane
contains its chord-line, which would otherwise be inclined thereto.
With this change the formulae (ii) and (iii) of Article 128 become
for points on the boundaries of cambered aerofoils —

X + i Y = (1 + i tan 2(J)/(X1 + iYJ . (iiA)

VI]

and —

TWO-DIMENSIONAL AEROFOILS

. 227

(iiiA)

X = (Xl + Yl tan

Y - - (Yl - Xl tan 2P)//?/ '

whilst to relate Xlt Yl to (ji for points on the hyperbolic boundary
we now have in place of (iv) of Article 128 —

e + cosh (ji cos T/2

= tan

(ivA)

cos

Thus the only modification of the values of Xv Yl for points on
the hyperbolic boundary is the inclusion of tan 2(3 in the expression
for Y1. It follows that the relation (v) of Article 128 between Xl
and Yl will be applicable to cambered aerofoils of the family if Yl
is replaced by Yl — tan 2(3.

Fig. 96B shows two cambered aerofoils of this family.

LIFT OF WINGS OF INFINITE SPAN
134. Joukowski's Hypothesis

Suitable values being chosen for parameters and a definite aerofoil
shape obtained in the tf-plane, the same transformation process con-
verts the streamlines past the circle to the corresponding stream-
lines past the aerofoil. Now when this process is applied to flow
without circulation in the z-plane, results follow of which Fig. 97

FIG. 97. — STREAMLINES PAST JOUKOWSKI AEROFOIL WITHOUT CIRCULATION.

is typical ; the back stagnation point, 5, on the circle transforms
to the back stagnation point S' lying on the upper surface of the
aerofoil some distance in front of its trailing edge. As with the thin
plate, normal or inclined to a stream, fluid is asked to whip round a
sharp edge, attaining an infinite velocity in the process.

228 AERODYNAMICS [CH

It is easily proved, as follows, that for all conformal transforma-
tions the circulation round the aerofoil is the same as the circulation
round the circle. Construct any two corresponding circuits enclosing
the circle and aerofoil respectively. Then, since <f> is the same at
corresponding points (Article 122), the interval of ^ round each
circuit will be the same. But the circulation is the interval of <f>
round a complete circuit. It is important to note that this result
is independent of the relationship between the undisturbed velocities
in the two planes.

In Fig. 97, therefore, there is no circulation round the aerofoil,
and it will shortly be proved generally that no force arises on the
aerofoil in these circumstances. Now, the criticism that fluid cannot
turn round the sharp trailing edge might be met by rounding that
edge, which could be achieved by enclosing all singular points within
the aerofoil, as we have seen. But the result of zero lift, incompa-
tible with experiment, would still suggest the streamlines to be
discordant with fact.

Modifying the streamlines past the circle by adding a small
circulation K displaces the point S' backward, and a particular
relationship between K and the undisturbed velocity makes S'
coincide with the sharp trailing edge, so that the velocity there
becomes finite.

Joukowski's hypothesis is that K is correctly and uniquely deter-
mined by the above consideration. Briefly, let Q be the point on
the circle which transforms to Q' the trailing edge of the aerofoil.
Since dt/dz == O at Q and

clearly only one condition permits of a finite velocity at Q', viz.
when q = 0 at Q, i.e. the value of K to be added to the flow past
the circle must be such as to make S coincide with Q. It is appli-
cable only to wings with a sharp trailing edge, although a tail angle
may exist. But it may be supposed that if we determine K for
such a wing, and then slightly round the trailing edge for ease of
construction, the effect of the modification will be small.

135. Calculation of K

Denote the undisturbed velocity by qot and let it be inclined at an
angle a to the #-axis. With K = 0, the stagnation points on the
circle are Slt S (Fig. 98). These approach one another in that half

VI]

TWO-DIMENSIONAL AEROFOILS

229

FIG. 98.

of the circle which transforms to the lower surface of the aerofoil
when K is added in the direction shown.

Referred to axes Bxlt Byl through B as origin parallel and per-
pendicular to q0t the flow past the circle is given by —

(i)

whence —

K
8 + .

<!• = - J. (r - -') sin 6 - ^ log r. . . (168)

If qb denote the peripheral velocity round the circle —

[3tin
£L - "ft

giving—

qb = o when K — — 471^0 sin 6

(Cf. Article 109).
Hence, for the stagnation points to recede to Q, S, —

K = 4nbq9 sin Y

= 4nbqQ sin (a + p) . . . (169)

from the figure, determining K in the z-plane.

The figure refers particularly to the Joukowski transformation,
but the theorem is general. When less simple transformations are
employed, however, care must be taken to note that the velocity

230

AERODYNAMICS

[CH.

and angles of (169) refer to the circle-plane and may be changed
in passing to the aerofoil-plane.

136. The Streamlines

Plotting (168) with the prescribed value of K/q0 gives the stream-
lines appropriate to a chosen value of a in the 2-plane (cf. Article
109). Transforming these gives the flow past the aerofoil. An
example is shown in Fig. 99. The value of K/q0 and, therefore, the

FIG. 99.— STREAMLINES PAST THE AEROFOIL OF FIG. 97 WITH CIRCULATION
ACCORDING TO JOUKOWSKl'S HYPOTHESIS.

streamlines past the aerofoil, will change if a be varied. Thus the
method is generalised as regards angle of incidence of the aerofoil.

In the 2-plane there is a lift L per unit length of the circular
cylinder given by —

L = pj^0. .... (170)

This force is perpendicular to the direction of q0.

The velocity round the profile of the aerofoil may be obtained
by the methods already described, and hence, from Bernoulli's
equation, the variation of pressure. Finally, the lift may be
evaluated by a graphical integration (cf. Article 44), and will be
found to be the same as L for the same undisturbed velocity.

Analytical investigation is given in the following articles.

137. The Lift

In the *-plane draw a circle of large radius R enclosing the aero-
foil at its centre (Fig. 100), and take axes 05', OTJ' parallel and
perpendicular to q<>. Since R is great, the circulation velocity com-
ponent at the circle is unaffected by the shape of the aerofoil (cf.
Article 115). It equals K/2-xR, and is perpendicular to R. At any
point P (R, 0) it has components cos 8 . K/2rcR perpendicular to q<>

VI]

TWO-DIMENSIONAL AEROFOILS

231

FIG. 100.

and sin 6 . K/2nR parallel to q0. Thus the resultant velocity q at P
is given by —

K V / K

Consider an element of the circle J?.S0 at P, and let m be the fluid
mass crossing it per second. We have —

m = pg0 . R8Q . cos 0. . . (ii)

When P is on the upstream side of the aerofoil, the streamlines
having an upward trend, passage across the element communicates
upward momentum at the rate m cos Q.K/2nR to the fluid within the
circle. This calculation is correct wherever the element is situated.
Hence the fluid within the circle will, on account of the flux of fluid
across its whole contour, have its momentum in the direction OY)'
increased at the rate —

""«* we

We have omitted to attach a sign to q0, and it is evident that this
should be negative, since the velocity is in the direction — 0%'.
Hence the last member of (iii) when essentially positive gives the rate
at which the fluid within the circle is receiving momentum from the
aerofoil in a downward direction, i.e. in the direction — OT\ '. This

232 AERODYNAMICS [CH.

is checked by the fact that the aerofoil bends the streamlines down-
ward.

The fluid outside the circle exerts, we shall also find, an upward
force on the fluid within by virtue of the pressure p acting radially
inward. This must also be taken into account.

Considering again the contour-element R8Q, the upward force on
it is — p sin 6 . R8Q. Integrating round the circle we find the whole
force to amount to —

f2T

Now p is related to q by Bernoulli's equation. If pQ is the undis-
turbed pressure of the stream, using (i) —

and since R is large, the velocity term in I/R* is negligible compared
with that in l/R, so that —

Substituting in (iv) we find for the upward force on the fluid within
the circle —

P;r-(

(v)

Summing up, we find that the fluid within the circle receives
downward momentum at the rate %pKqQ, while also it presses down-
wardly on the surrounding fluid with a force of the same magnitude.
Hence the upward reaction L on the aerofoil in the tf-plane is given by
(170). It is important to remember that by ' upward ' is meant
the direction 07]' which is perpendicular to that of qQ.

The equality of the momentum and pressure integrals in the fore-
going has no physical significance, following only from choosing a
circle for ease of integration. Variations are dealt with in Tietjen's
Applied Hydro- and Aeromechanics. A wing flying through the
atmosphere must derive its lift eventually from a pressure integral
over the ground or sea. This must amount to the same as the lift
calculated above.

138. The important result of the preceding article does not
depend upon the precise shape of the aerofoil. For aerofoils of the
simple Joukowski type we may substitute from (169) in (170),
obtaining —

L = 4p£0* . 7c6 sin (a + P) • . . (171)

VI] TWO-DIMENSIONAL AEROFOILS 233

Introducing the lift coefficient CL, and remembering that b/a~l+mf
CL 5

= 2w(a + p) . . . . (172)
approximately, when a, p, and m are small. In these circumstances

Tsr1— ..... <>">

i.e. if the angle of incidence of a Joukowski aerofoil of infinite span
increase, the lift coefficient CL increases at the rate 2n per radian, or
(HI per degree.

139. Pitching Moment

The most general transformation formula by which the flow past
aerofoil shapes may be derived from that past the circle is of the
type—

C' C"

* = z + -, - + ^ + ......

z z
where the coefficients are complex numbers. This gives

and all the zeros, except that yielding the sharp trailing edge of
the aerofoil, must be enclosed within the circle. The origin 0 is
situated at the centroid of the zeros. Different sets of poles and
circles may be chosen to give an infinite variety of aerofoil shapes.
Further development of this wider view of a subject of considerable
practical importance is left to subsequent reading and research.

The pitching moment about any point exerted by the pressures on
a given aerofoil can be determined as an application of the process
described in Article 136. General analytical investigation may
proceed as follows :

Consider a great circle of radius R with centre at 0. The pressures
round it are everywhere radially directed and exert no moment on the
fluid within. The pitching moment M9 about 0 can be calculated
from the rate of change of the moment of momentum of the fluid
passing through. If the resultant velocity q at the point R, 8 is
inclined at e to 0%, the mass of fluid crossing the element RSQ per
second is p<? . cos(e — 0)jRS6, while its velocity perpendicular to R
is q. sin(e — 0), and the moment of its momentum is accordingly
sin 2(e — 6)J?280. Integrating round the circle —

J o

sin 2(c — eye = o.

A.D.— 8*

234 AERODYNAMICS [CH.

Now from (136), Article 110, if u and v are the velocity components
in the /-plane of the aerofoil —

dw / • • \ *

— = u — iv = #(cos e — i sin e) = qe

whence —

—Mo = foR* * ( -^)f ^ sin 2(e — 6)^8 . (ii)

and the problem is resolved into finding a tractable expression for q
with reference to the axes 0%, Ot] of the great circle.

The flow round the 6-circle is given in Article 135 (i), referred to
a Zj-plane, whose origin is at JS, and whose axes are at the inclination
a. This is transferred to axes through 0 parallel to 0£, OTJ by the
substitution —

*i = (z — z») e** . . . (iii)
where z0 is the co-ordinate of B, and becomes —

^ = — ?c

whence —
dw

Now-

rfze^ rf^ dz

~dt ~~~~dz 'Tt

and on expanding (iv) in descending powers of z and making use of
(i) we find —

The integral in (ii) can be solved * after substitution from (v), with
the result that MQ comes to the imaginary part of the expression—

where L is the lift. The second term represents the moment of the
lift acting at B about the origin 0. Omitting this, and writing
ipgft for C', we obtain for the moment MB about B —

MB = 27rp£0*A2 sin 2(oc + y) . . . (175)

This result is quite general. As an example, it may be shown that

for zero travel of the centre of pressure, a problem of practical

importance, particularly in connection with the structural design of

* Mrs. Glauert, loc. cit., p. 184. This proof is due to v. Mises. A different treat-
ment is given by H. Glauert, The Elements of Aerofoil and Airscrew Theory, Chap. VII.

VI] TWO-DIMENSIONAL AEROFOILS 236

wings, we must have y = p, which circumscribes the form of C' in
(174). To see this, we note that for a fixed C.P. when drag is nil, the
moment must vanish when the lift vanishes, and that the latter
occurs at the incidence — p.

If we now restrict the result to the simple Joukowski transforma-
tion formula, so that C' = af, y vanishes and —

MB = 27cpj0f0a sin 2a. (176)

The moment coefficient Cm is sometimes defined from the moment
M about the leading edge of the aerofoil. Since the leading edge is
distant 2a from B, it is then given closely by —

M = M - 2aL

from (172), where c is the chord = 40. Hence, 5 being the area, = c
per unit of span —

2oc *

8 2

= -2P~4CL '
approximately.

140. Comparison with Experiment

Comparisons with experiment of practical engineering interest will
occur in Chapter VIII, after extension of the theory to three-dimen-
sional aeroplane wings. Numerous successful checks on the theory
at the present stage have been devised, however, by experiments
arranged to imitate two-dimensional conditions of flow. Such
investigations are easily carried out and form interesting laboratory
work. A long aerofoil is made to a Joukowski section which has
been worked out in detail on the drawing board, and is mounted to
stretch between the walls of an enclosed-type tunnel or right through
an open jet. Preferably it is carried on traversing gear, so that the
velocity at a single point in the stream can be measured in direction
and magnitude with the aerofoil in various relative positions. It is
also fitted for measurements of normal pressure round its median
section, or sections close thereto.

The fundamental conception that lift, per unit of span, = $K x
velocity, is closely realised on assessing K by graphical determination
from measurements of the line integral of the tangential velocity
round any wide circuit enclosing the aerofoil and cutting the wake
roughly at right angles. For circuits that approach the aerofoil

236 AERODYNAMICS [CH. VI

closely (without, of course, cutting the boundary layer) K may
decrease by some 10 per cent. The lift is determined for purposes of
comparison from the experimental pressure diagram, as already
described.

On examination, the pressure diagram will be found to conform
reasonably closely with that determined theoretically for the section
and incidence. Observations tend, however, to lie within the calcu-
lated diagram, differences occurring chiefly near the crest and tail
of the upper surface of the section. Thus the experimental lift is less
than the theoretical, although it is in agreement with the observed
circulation ; the theory over-estimates what a given shape can do,
owing to neglect of frictional effects. If the pressures be observed at
various small incidences and suitably integrated, it will be found that
the slope of the lift coefficient curve is less than 2?r. The value 6 is
often used instead, although even this value is too generous and 5|
is much closer to fact. For incidences approaching the critical
angle the theory completely breaks down.

Typical pressure diagrams are not illustrated, since they resemble
those already given for the median sections of aerofoils of considerable
aspect ratio. The essential difference is that, in the case of two-
dimensional aerofoils whose camber and thickness ratio are small,
the pressure drag becomes small at moderately high Reynolds
numbers, whilst in the case of the median or other planes of a
three-dimensional aerofoil it does not do so, though the section be
the same, unless the incidence is such that the lift is also small.

Chapter VI A
THIN AEROFOILS AT ORDINARY SPEEDS

1 40 A. The preceding chapter gives an introductory account of
a method by which the potential flow of an incompressible fluid
past aerofoil sections resembling those of aeroplane wings can be
obtained accurately and without difficulty. The fully developed
theory provides for some 9 parameters controlling the shape of the
profile, so that given wing sections can if desired be fitted closely.
Arbitrary shapes can also be dealt with by an approximate method
due to Theodorsen, further developed by Goldstein. Again, the
potential flow problem could be solved otherwise by determining
an appropriate distribution of vorticity round the given boundary.
For it can be proved * that every continuous irrotational motion of
an incompressible fluid that extends to infinity and is at rest there
may be regarded as due to a certain distribution of vorticity round
the surface of the body producing the motion. The determination
of the said distribution in a given case involves in general, however,
an integral equation whose solution is laborious without special
calculating machines, and therefore this line of approach is followed
in this book only in the present chapter.

Consideration of cambered aerofoils has so far been limited to
those whose camber-line is a circular arc, a shape that is unfavour-
able for practical use since it entails a large moment coefficient,
cf. (177). There is no analytical need for this restriction, and
development of the system of conformal transformation described
in Articles 128-129A and 133A enables the camber-line to be
varied in shape as desired without loss of ultimate accuracy. Such
variation has several applications and particularly to the reduction
of the pitching moment coefficient. When, for example, the crest
of an unreflexed camber-line is advanced from the mid-chord point
characterising the circular arc to a position one-sixth of the chord
behind the nose, the moment at zero lift almost vanishes and the
centre of pressure remains almost stationary at the quarter-chord
point. So far forward a position for the crest of the camber-line is

* Lamb, Hydrodynamics, 6th Ed., p. 214.
237

238 AERODYNAMICS [CH.

often unacceptable for other reasons, but evidently some advance
from the mid-chord point is important. Similarly, Piper* has
extended the simple profiles of Article 133A so that a stationary
centre of pressure is secured by reflexing the camber-line towards
the tail and displacing its crest upstream. These and similar
precise calculations accord with experimental results known for
some time previously. But they also show f that for small cambers
and thickness ratios the effect of the distribution of thickness of
the section on the pitching moment is small compared with that of
the shape of the camber-line. Hence a serviceable approximation
to the moment should be obtainable in such cases by neglecting
thickness altogether and regarding the aerofoil simply as a bent
line, the median line of its upper and lower surfaces. Such a
' skeleton ' is intended (in the present connection only) by the
term ' thin aerofoil/ Thus the circular arc of Fig. 93, if restricted
to a small camber, constitutes the thin aerofoil (in the present
connection) for all the transformations considered in detail in
Chapter VI. This may be thickened into a Joukowski section or
into one whose profile is inverted from an hyperbola, and the
moment at zero lift will be different in the two cases but negligibly
so, compared with the difference that would result from changing
the camber-line if both thickness and camber are small.

The bent line to which the aerofoil section is reduced may be
regarded as the trace of a surface of discontinuity in the sense that
the velocity of the fluid, though tangential to it at all points,
differs in magnitude at adjacent points on the two sides of the line.
This difference is determined by a distribution along the line of
' log r sources/ or ' point vortices/ i.e. the simple element-motion
described in Article 103 without restriction as to the minimum
value of r except as stated in Article 39. The total circulation K
round a circuit enclosing the line is equal to the sum of the circula-
tions of all these elements (Article 97). Determining this by
Joukbwski's Hypothesis secures a finite velocity at the trailing
edge but, since the front stagnation point is under the sharp nose,
the velocity is infinite at the leading edge. A further approxima-
tion, as follows, is also introduced at the outset. Investigation
being limited to small cambers, i.e. to small deviations of the
camber-line from the chord-line of the section, it is assumed that
the vorticity may, for purposes of calculation, be regarded as
distributed along the chord.

* Loc. cit., p. 212.

t Garrard, Ph.D. thesis, London, 1042.

VIA]

THIN AEROFOILS AT ORDINARY SPEEDS

239

1406. The Equations

Choose the origin midway along the chord ; draw Ox along the
chord-line in the upstream direction ; and let the undisturbed
velocity V make an angle a with Ox. Denote the chord by ct and
let it be understood that the following integrations with respect to
x are to extend from x = — \c to x = + \c.

Let k be the local intensity of the circulatory function forming the
surface of discontinuity, so that —

= f MX.

(i)

The lift L per unit of span is

L =

kdx

and, the local contribution to lift being proportional to kt the
moment M about the mid-chord point is given by —

M = PF I kxdx. . . (iii)

Kdx

The contribution, 8^, of the element kdx to the normal velocity
component v1 at x± (see Fig. 100A) = — kdx/2n(x — x^, whence —

kdx

_ __L f

Vl ~~ 2n J

X — X*

(iv)

This velocity, though determined for the point x± on the chord, will
be approximately the same at the corresponding point P on the
thin aerofoil. But the resultant of vl and V must be parallel to
the tangent at P. Now this tangent is inclined to V at the angle
a _|_ dyfdx, so that the normal component of F at P is F(a + dy/dx).
Thus the boundary condition requires —

F(a + dy/dx) + v = 0,

240 AERODYNAMICS [CH.

or by (iv) —

f MX = ft | dy ^ ^ (y)
7 j x — xl dxt

and this equation is to be satisfied for all points on the thin aerofoil.
Thus the complete solution of the problem follows the determination
of the distribution of k along the chord which will satisfy (v) in
respect of a specified thin aerofoil defined, in the present connection,
by the variation of the slope dy/dx of the camber-line along the
chord.

I40C. Application to Circular Arc

Before attacking directly the problem presented by (v) of the
preceding article, it is useful to investigate a case for which the
solution is already known, viz. the circular-arc thin aerofoil of
Article 130.

Generally, the assumption of small camber implies that the
difference in the tangential velocities at the upper and lower surfaces
of the thin aerofoil at a given point can arise only from the vorticity
at that point, since other elements produce only normal velocities
there. Hence k is proportional to this velocity difference, which is
readily evaluated, as follows, for the case chosen.

Referring to Fig. 93 the circle with centre at B and passing through
the two singular points Q and R is transformed by Joukowski's
formula into the two sides of a circular arc of camber jp. In the
nomenclature of the figure, let the chord Q'R' be at a positive in-
cidence a to a stream of velocity V coming from the right. With a
circulation appropriate to Joukowski's Hypothesis, the velocity at
any point Pon the circular boundary is obtained from Article 135 as —

ft = 2F[sin (G! + a) + sin (a + P)],

where 6t is the angle that BP makes with the x-axis. Or approxi-
mately, for a and p small,

?I = 2F[sin 6X + oc(cos 8t + 1) + p]. . (i)

Now by Articles 132-3, to the first order, for a Joukowski aerofoil
of vanishing thickness r/b = r/a = 1 + p sin 6 and x/c = % cos 6,
Hence, and from the figure,

sin G! = (r sin 6 — a$)/b = sin 6 — p cos1 6,
cos 6X = (r cos 8)/6 = cos 0 + p sin 6 cos 0.

Substituting in (i) gives —

ft = 2F[sin 6(1 + P sin 0) + a(l + cos 6)], . (ii)

VIA] THIN AEROFOILS AT ORDINARY SPEEDS 241

squares and products of the small quantities a and p being neglected.
To the same order the modulus of the transformation simplifies
to—

= 2 sin 6(1 — p sin 6),

so that the velocity on the boundary in the aerofoil plane is given
by-

Q = ~-J^ = vri±l!E® . i + cos e i

q* I dt/dz | L! - P sin 6 "^ sin 6(1 - p sin 6)J

- V \ I + 2p sin 6 + a AJ^L6] . . . (iii)
L sm 6 J

for a and p small, and the last term can be reduced.

Now if for any point on the upper surface of the circular-arc
aerofoil the corresponding point in the circle plane subtends the
angle 0, the adjacent point on the lower surface has a corresponding
point subtending the angle —0. Hence the difference between the
increased velocity at the first point on the aerofoil and the reduced
velocity at the second point is equal to —

2F[2p sin 8 + a cot (6/2)], . . (iv)

to which k is proportional. The first term arises from the camber
and the second from the incidence. Only the second would be
present in the case of an inclined flat plate.

It will be noticed that at the trailing edge, defined by 6 == n,
the value of k is zero. This result is quite general, arising from the
application of Joukowski's Hypothesis, for the velocity could not
otherwise remain finite at the trailing edge.

1400. The Genera] Case

The above result suggests that for more complicated thin aerofoils
than the circular arc a suitable assumption regarding the variation
of k along the chord is —

k = 2tcF L40 cot - + ItA n sin w8j,

where 6 is now a variable related to the distance x measured along
the chord from its mid-point by —

X = \C COS 9,

242 AERODYNAMICS [CH.

and only integral values of n are considered in the summation, so
that the condition k = 0 at the trailing edge remains satisfied
however many terms are included.

Substituting this assumed form for k in equation (v) of Article
140B gives the following expression for the slope dy/dx of the thin
aerofoil, or camber-line —

T +
dx

•-f

AQ COt - +

cos 6 — Cos

sin 0^0.

(i)

The evaluation of the integral gives rise to some difficulty owing
to the singularity at 6 = 0P where the denominator vanishes.
We have to obtain the so-called principal value by evaluating the
integral in two parts, between the limits 0 to 0X — • e and 0X + e
to TT, and then evaluating the limit as e becomes vanishingly small.

The principal values of the integral —

are*

••-J.5

cos nQ

OS 0 — COS 0!

> 1

sin nQt

sin 0X

With the help of these, (i) can be re-expressed and reduced as
follows : —

AQ(l + cos 0) + $XAn[cos (n - 1)0 - cos (n + 1)0] M

cos 0 — cos 0X
sin (n + 1)0! — sin (n —

sin Ui
= TU {A0 — TtAn cos nOj}. .... (ii)

The values of the coefficients may be evaluated directly if the
slope of the thin aerofoil can be expressed as a sum of cosines in
this form. Alternatively, they may be found by the usual pro-

* Cl, for instance, Glauert, Aerofoil and Airscrew Theory.

VI A] THIN AEROFOILS AT ORDINARY SPEEDS 243

cedure employed in the calculation of the coefficients of a Fourier
series. Dropping suffix 1 as no longer necessary and integrating —

cos

J4oE. The Aerodynamical Coefficients and Centre

Equations (ii) and (iii) of Article 140B are now solved. Expressed
in terms of coefficients, they give the following —

2/^ f77

CL = — = 2n\ [A0(l + cos 6) + 2AMsin nQ sin Q]dQ

V C Jo

- 27u'(,40 + i^). . . . • . . (i)
Denoting by CM' the moment coefficient about the mid-chord point,

CM' =7c| [A0(l + cos 6) + 5L4n sin nQ sin 6] cos 6 dQ
Jo

+ i^f) ...... (ii)

If CM without an accent denotes the moment coefficient about a
point one-quarter of the chord from the leading edge,
CM = CM JCL

= ^(4,-^) ...... (iii)

The last result is independent of incidence since only AQ involves
a. The quarter-chord point is therefore called the Aerodynamic
centre. The value of CM is a measure of the centre of pressure
travel sin£e the distance of the centre of pressure from the quarter-
chord point is CM/CL. It will also be noticed from (iii) that a fixed
centre of pressure requires that A2 = Av

i4oF. Example
Let us assume for the camber-line a cubic curve of the form —

lc = (J - #!<*) (A + BXJC),

so that —

^ = - *(*lc)(A + Bxfc)

= iB - 2Ax/c — 3Bx*/c*,
or, in terms of cos 0,

dv 1 3

-f = — ~B — ^ cos 0 -- B cos 29.

dx 8 8

244

This gives

A, =

AERODYNAMICS

.
Sn

[Cn. vi A

The condition for a fixed centre of pressure is thus B = 8/4/3. We
conclude that a camber-line of the type concerned will secure zero
travel of the centre of pressure for a thin aerofoil provided its
equation is —

-• +

where C is a small coefficient. This shape of camber-line is illus-
trated in Fig. 100B, showing the magnitude of the reflexure towards
the tail.

FIG. 100B.

Chapter VI B
COMPRESSIBLE INVISCID FLOW

1406. In Chapters V, VI, and VI A, the motion of the inviscid
fluid considered was assumed to be irrotational and incompressible,
steady and two-dimensional. The velocity potential that exists
under these conditions satisfies Laplace's equation, making rapid
progress possible, and the assumptions will often be reintroduced
to obtain approximate solutions to practical problems. But in
some circumstances they are inadmissible ; examples occur when
high speeds cause appreciable variations of density, or when three-
dimensional concentrations of vorticity exist in otherwise irrotational
flow ; in the latter case there is no ' stationary ' streamline state,
and we have to face unsteady as well as three-dimensional conditions.

The present chapter is concerned in the first place, therefore,
with establishing a broader basis of fundamentals for guidance in
later investigations. The fluid is still regarded as inviscid, so that
the pressure acts equally in all directions at any point and there is
no diffusion of any vorticity present. It is also assumed that the
flow proceeds isentropically without shock (Chapter III), by which
is meant generally a sudden large increase of density and decrease
of velocity associated with a production of vorticity and other
phenomena. In the succeeding chapter the general theory will be
applied to two-dimensional aerofoils at high speeds.

I40H. Generalised Equation of Continuity

Three-dimensional motions will be referred to fixed co-ordinate
axes Ox, Oy, Oz, in the direction of which the velocity components
at any instant are u, v, w. The equation of continuity for unsteady
compressible flow depends on little more than the conservation of
matter ; no special form is obtained for an inviscid fluid. Fixing
attention on a box-like element of space, of sides 8#, Sy, 8z, the mass
enclosed at any instant is p . 8#8y8z, and at the end of a brief interval
of time $t this mass is increased by —

(i)

246

246 AERODYNAMICS [CH.

Let the centre of the box be at the point (x, y, z) where, at any
instant, the velocity components are u, vt w and the density is p.
The mass entering the box during the time & through the ^>
nearer to the origin is —

while the mass leaving the box through the opposite face is

In respect of flow through this pair of faces, therefore, the mass of
fluid within the box increases during 8t by the amount —

-
dx

Extending the calculation to include also the other two pairs of
faces, the total increase of mass during the time is —

But this must be equal to (i). Hence —

!+£+£+£-«•• • <«>

This general equation of continuity for compressible flow can take
various equivalent forms. Simplified for steady two-dimensional
flow to compare with (61), it gives —

du dv I I dp 3p\

= I u ~ + v ~ } , . (in)

dx dy p\ dx dyJ v '

showing a great change from the equation for incompressible flow.

140!. It is frequently required to know the component accelera-
tions of a fluid element, and brief consideration shows that formulae
for this purpose can be constructed only by following the element
for an instant along its path ; a simple example has already arisen
in Article 29. The same position arises if it is desired to calculate
the rate at which some property of the element other than its
velocity is varying. We therefore examine the rate at which any
function of position and time, f(x, y, z, t), varies for a moving
element.

If at time t a particle occupies the position (x, y, z)t at time

VI B] COMPRESSIBLE INVISCID FLOW 247

n (

t + 8t it will have moved to the position (x + 8x, y + 8y, z +
and the function will have increased to —

J dx dy dz dt

Writing the new value of the function in the form

Df _ a/ a/ a/ a/
Di~dt + ud~x + vty + wdz-

The operator Z>/Dtf, defined by —

is sometimes known after Stokes, and its use is called a differentiation
following the motion of the fluid. It is worth remembering.

For example, the rate at which the density of a moving element
is increasing is —

Dp 3p 3p 9p Sp

~W+==zt+uz+v^ + w~^'
Dt dt dx dy dz

Substituting in (ii) of the preceding article gives —
Dp fdu dv d

providing an alternative, and often more convenient, form of the
general equation of continuity.

140). Euler's Dynamical Equations

From the preceding article, the component accelerations of an
element are —

l)t

= ¥H

h"ai +

V^H

~WVz

a«

V-w^

~Dt

- ¥H

r M"^~ ~h
dx

vty~

'dw

— — — — — j

U M -4-

. mi _

L- ju _

dt ^

h 3;c

1 w />

The force components producing these accelerations are, in the
absence of viscosity, only of two kinds. The first arise from the
pressure gradients, as described for one-dimensional motion in

248 AERODYNAMICS [CH.

Article 28, and are proportional to the volume of the element.
The second are due to extraneous causes, such as gravity.

At any instant let p be the pressure, p the density, and X, Yt Z
the components of extraneous force per unit mass at the point
(x, y, z), and consider an element of volume V with its centre at
this point. Resolving in the ^-direction, pF . Du/Dt = pF . X —
V . dp/dx and similar expressions result from resolving in the y-
and z-directions. Hence finally,

Du
~Dt

^X-^P

Dv
Dt
Dw

pdx

= Y - i^

(ii)

I dp

p dz'

These equations of motion are general, the only restriction being
to an in viscid fluid. Substitutions for the left-hand sides will be
made from (i) in accordance with given. conditions ; e.g. the terms in
d/dt will be omitted if the flow is steady, and those involving w if it
is two-dimensional.

I40K. Kelvin's (or Thomson's) Theorem

It will now be shown that under certain conditions the circulation
round any circuit moving at every point with the fluid does not
vary with time. This theorem was enunciated by Lord Kelvin
when Sir William Thomson and is known under both names. It is

of outstanding import-
ance in Aerodynamics,
exercising a directive in-
fluence on the theory and
design of wings, air-
screws, wind tunnels, etc.
The above conditions are
as follows : (a) The fluid
is inviscid, though it may
have any distribution of
vorticity and is not con-
strained to be in steady
motion ; (b) There exists
an integrable functional
FIG. looc. relation between the

VI B] COMPRESSIBLE INVISCID FLOW 249

pressure and density ; (c) The extraneous forces are con-
servative.

Consider first any two separated points, A and J3, Fig. lOOc,
themselves moving with the fluid and connected by any ' fluid
line/ i.e. a line of which every point is moving with the fluid ; thus
the line selected will always consist of the same fluid particles,
however its shape may vary with time.

The rate at which the flow along the fluid line from A to B
increases is given by —

D rB

— I (udx + vdy + wdz). . . (i)

Considering the first term of this expression,
D . Du .

_l = _ ,._.

But (D/Dt)(8x) is the rate at which the projection of an element 8s of
the line on the #-axis is elongating, i.e. it is equal to Su. Hence —

~t (u$x) =D~** + «Sw = D£t*x + !«(«•),
or, substituting from the equations of motion,

Differentiations following the motion of vlly and w$z are similarly
expressed, yielding three such equations. It is assumed that the
components of extraneous force per unit mass are derivable from
a single- valued potential function ii, so that X = d£l/dx, etc.
Hence, adding the three equations together,

p \ dx dy J dz

t/« -f «/«)

= 80 - +
P

Integrating along the entire fluid line gives finally—

[cdfi
ft — I — + i?1
j p

250 AERODYNAMICS [CH.

Now let the fluid line be elongated to form a closed circuit, A
and B becoming adjacent points. The left-hand side of (ii) then
gives the rate at which the circulation K round this circuit varies
with time. But the value of the right-hand side must be zero
when B coincides with A, provided p is an integrable function of p,
as assumed. Hence —

DK „

-^-=0, . . . (ni)

i.e. the circulation round a loop of fluid particles remains constant
provided it does not enter a rotational field of extraneous force and
the chain remains unbroken.

Applications of this result will be discussed in place, but the
following may be noted at once. If the motion of a bulk of inviscid
fluid is at any instant irrotational, then the circulations round all
fluid loops that can be drawn within the bulk vanish for the reason
discussed in Article 97. The theorem then asserts that these
circulations remain zero under the conditions assumed, i.e. that the
state of irrotational motion is maintained in that bulk of fluid.

IRROTATIONAL COMPRESSIBLE FLOW
i4oL. If at any instant all elements of the fluid (or of a given
part of a fluid in motion, steady or not) are devoid of vorticity, it
can be shown as before that a velocity potential then exists, since
every closed circuit that can be drawn in the region occupied has
zero circulation. But the argument can be shortened to the
following.

If at any instant udx + vdy + wdz is an exact differential d<f>,
then u = d<f>/dx, v — d(f>/dy, w = d<j>/dz, and it follows immediately
that —

dw dv __ Su dw __ dv du

ty-~dz = °' dz " Tx = °' dx ~~ dy = °' W

The left-hand sides of these expressions are the components of
vorticity of an element situated at (x, yt z). Assuming (i) in the
first instance establishes the existence of a velocity potential at
the given instant, irrespective of compressibility. The theorem of
the preceding article then enables us to say that the bulk of fluid
considered will continue to possess a velocity potential.

I40M. Integration of the Equations of Motion

Euler's dynamical equations can be integrated through any part
of a fluid in which a velocity potential exists. The need for this

VI B] COMPRESSIBLE INVISCID FLOW 2fil

step is not always apparent on a first reading of the subject in view
of Articles 29 and 40. But it will be reflected that Bernoulli's
equation was derived by integrating along a streamline, whilst in
potential motions surrounding concentrations of vorticity, for
example, streamlines in the sense implied do not in general exist,
but only path-lines.

By virtue of (i) of the preceding article, the first of the equations
of motion, viz. —

du du du du „ 1 3/>

a7 + u a" + VT- + w a -- X + " *' = °
ot ox dy dz p dx

can be changed to —

d26 du dv dw „ 1 df>

a-75 + u^ + v *- + w a -- X + ~ / = °-
cxct dx dx dx p dx

Similarly, the second and third of the equations of motion can be
rearranged to involve partial differentiations with regard to only
y and z, respectively. Multiplying the first of these rearranged
expressions by 8x, the second by Sy, the third by 8z, and adding
gives —

8( &

But u8u — £8(^a), etc., and u* + v* + w* = q2. Hence, assuming
that X, Y, Z have a potential Q, say, so that X — dd/dx, etc., the
last equation becomes —

Integrating,

In general C is an arbitrary function of time, and is therefore more
accurately written as F(t), or absorbed into d</)/dt with this under-
standing and the left-hand side of the expression then equated to
zero. In this strict sense the left-hand side of (i) is constant for
all particles only at any instant ; its value can be altered by, for
example, changing the pressure throughout the bulk of fluid by
extraneous means, such as a pump. But Aerodynamical calcula-
tions usually suppose an unrestricted expanse of fluid and exclude
such external actions, when C becomes a constant.

252 AERODYNAMICS [CH.

For steady flow in the absence of extraneous forces the equation
(i), obtained by integrating Euler's equations of motion, reduces to— -

»-£- + %q* = constant, . . (ii)
P
i.e. to the same form as Bernoulli's equation, found by integrating

along a streamline, Article 29. The new result is less general in
that it is restricted to potential flow, but it is more general in
showing that with that type of compressible steady motion
Bernoulli's constant has a single value for all particles.

As before, the pressure is assumed to be related to the density by
the adiabatic law p = kpyt and expressions obtained in Articles 31
and 32 are applicable under steady conditions.

I40N. Steady Irrotational Flow in Two Dimensions

The equation of continuity for steady compressible flow in two
dimensions is, from Article 140H,

du dv __ 1 / 3p 9p\

If the flow is irrotational, this becomes on substituting for u and v
in terms of <f> —

where y1

Differentiating (ii) of the preceding article and remembering
that dpjd? = y^/p = aa,

dq* _ 2 dp 9p __ 2a* 3p

3^ p rfp 3# p 3^'

a?a _ 2aa 9p

8y p * dy'

where a is the local velocity of sound ; i.e. that for the air in the
region where the velocity is q. Substituting in (ii),

If the disturbed motion arises from a uniform stream of velocity
U and in which the pressure is pQ, the density p0 and the speed of
sound a0, then by Article 31 —

VI B] COMPRESSIBLE INVISCID FLOW 253

This expression enables substitution to be made in (iii) for the
variable local speed of sound, giving —

[2 - (Y - !)(«.' - M»)]V* = (O + W. (iv)

where qa is written for the velocity ratio q/aQ and M for the Mach
number U/aQ.

This differential equation for ^, the counterpart, for compressible
flow, of (115), has no general solution and is laborious to handle.
<f> is usually expanded in a series of terms of even powers of M, but
convergence becomes slow as q approaches a ; solutions * have been
published only for the circular and elliptic cylinders, a Joukowski
aerofoil and the sphere. A hodograph method, introduced by
Tchapliguine in Russia (1904) has recently been developed f in
the hope of enabling higher local speeds to be dealt with.

Whilst the exact calculation of compressible flow in two dimen-
sions for given boundary conditions is intricate even under favour-
able conditions, there is no difficulty in appreciating qualitatively
the effect of compressibility on the streamlines. As may be verified
directly or inferred from the preliminary discussion of a general
nature given in Chapter II, (iv) approximates closely to (115) for
moderately small values of the Mach number M = U/aQ. The
range depends upon the section of the body since the criterion is
associated with the maximum value of q/a attained by the fluid in
flowing past it ; thus it may be more than twice as great for a wing
section as it is for a circular cylinder. As M is increased, a stage
is reached, early or late, when the variation of density is no longer
negligible. The streamlines for incompressible flow are by then
appreciably distorted. Let Ss, $n be elements of length of adjacent
streamlines and equipotentials, respectively. For incompressible
flow, q = d(f>/ds = d^ldn. The first of these still holds for com-
pressible flow, but variation of p must now be taken into account
by defining 8fy = aq . oX where a denotes the density relative to
that of the undisturbed stream. Hence while formerly, with the
density constant, <f> and ^ varied equally through any small region,
their variations are now inversely proportional to a. Near the
stagnation point, where the density increases, the streamlines close
in, whilst near the shoulder of the body-section they separate

* Rayleigh (Lord), Phil. Mag., vol. 32, 1916. Hooker, A.R.C.R. & M., No. 1684.
1936. Imai and Aihara, Tokyo Univ. Rept., No. 199, 1940. Kaplan, N.A.C.A.T.N.,
No. 762, 1940. And others,

| Karman and Tsien, see the former, Jour. Aero. Set., vol. 8, 1941, where further
references are also given. The value of 7 is changed by this method.

264 AERODYNAMICS [CH.

farther from each other in order to accommodate between them
the same mass flow with a reduced density. The latter change is
usually the more important in Aerodynamical applications, and so
the main effect of compressibility is sometimes said to be an
expansion of scale across the stream, but this statement is in-
complete. Associated with the distortion of the streamlines, the
pressure changes round the profile of the body are augmented so
long as the flow remains irrotational.

1400. Analogies

These and similar considerations have led to the suggestion of
certain analogies with a view to inferring from convenient experi-
ments the effects of compressibility on irrotational flow. In an
electrical analogy, an alternating current is passed through a
layer of electrolytically conducting liquid contained in a bath
having an insulating bottom, which can be shaped to represent
the boundary condition in the flow case.* A trial exploration of
the distribution of electrical potential (which is proportional to <f>)
enables the distribution of a in the compressible flow to be assessed,
and the bottom of the bath is then re-shaped to make the thickness
of the electrolyte proportional to <r and the experiment repeated.
There has also been suggested an incomplete analogy with the flow
of water through an open channel,! as follows :

Hydraulic Analogy. — In comparing the two-dimensional flow of
a gas and the flow of an incompressible fluid, say water, with a free
surface along an open channel having vertical sides, an analogy will
be found to exist between the variation of the density p within the
compressible flow and the variation of the height h of the liquid
surface above the floor of the channel, which is assumed to be flat

- — - _ — — and horizontal. The water is assumed

to flow irrotationally and its velocity to
be constant at all points along any one
vertical line. Thus, if w is the com-
ponent of the (horizontal) velocity in
FIG IOOD " any direction across a vertical line whose

total height above the channel floor is h't

then the flux across a vertical strip of width b perpendicular to the
direction of w is w.h' b.

Let x be measured in the direction of mean flow in the channel
and y perpendicular thereto and horizontally, and consider a rect-

* Taylor (Sir Geoffrey) and Sharman, Proc. Roy. Soc., A, vol. 121, 1928.
| Jouguet, Jour. des. Math., 1920 ; see also Riaboushinsky, Pub. Sci. et Tech.,
Miaistre de Fair, No. 108, 1937.

VI B] COMPRESSIBLE INVISCID FLOW 255

angular space-element A BCD, Fig. 100D, whose centre is at #, y.
If the height of the layer of water is h at x,y, its heights at the middle

dh
points of AB, CD, are h ^ % — 8x, respectively, whilst the velocity

components perpendicular to these faces at their middle points are

Su
u T 1 TT SAC, respectively. Hence the rate at which fluid is leaving

the space-element in respect of flow across the faces AB and CD is,
by the foregoing,

- (« - &*} ("- 4» *+ (• + *f» (*+ *!«•) *

3* 9«

to the first order. Adding the rate of outward flow, similarly
calculated, across the pair of faces BC and DA, and expressing the
fact that the volume of the incompressible liquid within the space-
element cannot vary, we have finally —

dh T du dh , dv

u + h + v~~ + h~ =:Q
ox ox oy oy

i.e.,

Iw + ^w-o. . (i)

This result is identical with the equation of continuity for two-
dimensional compressible flow if p replaces h.

Relations derived by virtue of the absence of vorticity are
identical for the two cases of motion and, to establish the analogy,
it remains only to compare the relation of h to the resultant velocity
q in the channel, on the one hand, with the relation of the density p
to the resultant velocity in the corresponding compressible flow, on
the other hand. Suffix 0 will denote undisturbed conditions.

For the channel, Bernoulli's equation gives in the usual form
employed in hydraulics and which is easily deduced from Article
140M:

256 AERODYNAMICS [CH.

where pw is the density of the water. Applying this equation to the
free surface, where the pressures p and p0 are identical, yields —

But gA0 is equal to the square of the velocity of long waves of
small amplitude in a channel, so that putting ghQ = c* —

The corresponding result for compressible flow has already been
expressed in (47) as —

where a0 is the velocity of sound in the gas where the temperature
corresponds to p0.

Thus (ii) is identical to (iii) if h is substituted for p, c for aQ and if
y = 2. The value required for y is substantially different from
1-405, and the analogy is therefore incomplete in this connection.
This does not invalidate, however, its qualitative use. Moreover,
we observe that the incompleteness is negligible if q* — <?0a is small
compared with 200*, for then (iii) can be expanded as —

^o ~~ 2a<*~~ [2_ \ 2a02

If the density ratio is f , the term omitted in the analogy amounts
in this case only to about 2 per cent, for air. Thus the analogy is
close in the case of thin aerofoils and other slender bodies.

The formula quoted above for the velocity c of propagation of
long shallow gravity-waves is easily derived as follows.

Imagine such a wave travelling upstream and adjust the speed of
flow through the channel to c, so that the wave becomes stationary
and the entire motion steady. The flux per unit width of the
channel is then chQ.

Let h denote the height of any point P on the surface of the wave
above the bottom of the channel and q the fluid velocity at P.
Under the conditions postulated, the horizontal component of velo-
city u is sensibly the same at all points of the vertical line drawn from
P into the fluid and equal to q. Thus the flux across the transverse
plane through P is qh per unit width, and qh = chQ.

VI B] COMPRESSIBLE INVISCID FLOW 257

Bernoulli's equation gives for any streamline on the surface, the
pressure being constant,

fc* + gho = ka + gh>
Substituting for q,

a — 2$h*

This reduces approximately to the result stated, viz. ca = ghQt on
restricting the height ti of the crest of the wave so that A'//*o — 1 is
small.

The velocity a of pressure-waves in the atmosphere cannot be cal-
culated in this way without large error owing to the variation of
temperature consequent upon adiabatic expansions and compres-
sions. This matter has already been discussed and a reliable formula
given. The correspondence between the gravity-waves in a channel
and pressure- or sound-waves in air can be seen as follows.

Considering first a gravity-wave in slightly more detail, write
h0 + zl for h and let z denote height above the channel floor. Tak-
ing % in the direction of motion and assuming the disturbance to be
small, the pressure increase at the level z is approximately equal to
gfw (*o + *i — z)» the static value, whence dp fix = g^.dz^Jdx. This
is independent of z, so that every particle in a vertical line is displaced
equally.

Referring to the equations of motion given in Article 140J, all
other terms can be neglected in comparison with dujdt and dp fix ;
consequently —

du 1 dp dzl

dt pw dx dx

The upward velocity at P, viz. dzjdt, follows at once from the
equation of continuity for incompressible flow and the result that
du/dx is the same for all values of z since every particle on a vertical
line moves equally. We have —

3*,

Turning now to a plane wave of pressure disturbance moving
normal to itself in the ^-direction through the atmosphere, we
restrict investigation to the case of a small disturbance so that p/p0— 1
is small; this quantity is known as the condensation and denoted
by s.

A D. — 9

258 AERODYNAMICS [CH. VI B

With E written for the bulk elasticity pdpjdp and with udu/dx neg-
lected in comparison with Su/dt as before, the equation of motion is

du I dp E dp .

— •* — — \_ i vi i

dt 9dx P*a* * • v '

The equation of continuity for compressible flow must be employed
but, since the total variation in p is small (for which reason also a
suffix to distinguish undisturbed conditions is unnecessary), this
gives approximately —

1 9p du , ,.v

-P8i = -ai ' ' ' (V11)

Now equations (vi) and (vii) can be reproduced from (iv) and (v)
by substituting JE/p for gh and s for z^/h, establishing the analogy.
If the disturbance in pressure is other than small, we have to return
to the more complicated considerations set out for a plane shock
wave in Article 66D. The present simplification will be found to
possess, however, a surprising degree of utility.

In application to experiment, the floor of the channel will, of
course, slope slightly downward to counteract approximately the
effect of friction. The method has been widely employed, photo-
graphs being taken of flow patterns, especially of surface waves
above the critical speed ct forces on models being measured, and
surface configurations being used to estimate density ratios in the
corresponding compressible flows. Apart from these important
quantitative applications, the method is convenient to demonstrate
changes in flow which occur as the velocity is increased from well
below that of propagation of small waves in the medium to well
above this speed, corresponding to change from subsonic to super-
sonic flow. For example, the flow in a convergent-divergent
channel can be examined in this way and some aspects of the
supersonic tunnel revealed.

Chapter VI C
THIN AEROFOILS AT HIGH SPEEDS

I40P. Subsonic Speeds — Glauert's Theory

One application of conformal methods is found in the design of
sections for wings and airscrew blades intended for such high speeds
that account must be taken of the compressibility of the air. The
primary aim is to avoid the formation of shock waves by restricting
the maximum velocity ratio (cf. Article 129B) for a given lift
coefficient. The flow outside the boundary layer then remains
irrotational, as is assumed in the present Article. The difficulty in
the way of obtaining even an approximate solution of the exact
equation for <£ is avoided by deriving in the first instance an approxi-
mate form of that equation suitable for thin aerofoils. Owing to
the augmentation of pressure changes, modifications are required
to formulae (169)-(173) of Articles 136-138, and Glauert's Theory*
is directed towards establishing the basis for these. The theorem
proved in Article 137 for incompressible flow, and now written
for convenience as L = ^KU, still holds when the undisturbed
velocity U of the air stream is sufficiently high as to involve
appreciable variation of p in the neighbourhood of the aerofoil
from its initial value p0. This generalisation is assumed below. |

Investigation is restricted to thin aerofoils at small incidences
and having profiles that are everywhere inclined at only small angles
to the ^-direction of motion, so that the ^-component u of the
disturbed velocity is little greater than U and the ^-component
v is of the same order as u — U. The equation of continuity —

I (P«) + I (P*) = o, . . (i)
aw , dv if do ap

P P

i.e.,

dx dy p \ dx dy

may then be written approximately —

»» »

+ + _ o. . . (U)

ox dy p dx v '

* Pvoc. Roy. Soc., A, vol. 118, 1928.

f Proof is indicated by Taylor and Maccoll, Aerodynamic Theory, vol. Ill, Div. H.

250

260 AERODYNAMICS [CH.

Pressure and density variations are appropriately assumed to be
governed by the adiabatic law and, if 00 is the velocity of sound in
the undisturbed stream, (47) of Article 30 gives with the present
notation —

Differentiating with respect to x,

u du
I 3p __ aj dx

2a0' V ;

Now (y — l)/2 = 1/5 and (u 2— C/8)/a02 is always small. Therefore —
1 <5p __ _ u du __ __ U du

p dx aj dx a<f dx

closely. Writing M for the Mach number U/aQ and substituting in
(ii) gives approximately for the equation of continuity —

Substitution in (iii) of —

v' = v/(l - M2)1/s and y9 == y(l - M')1'2
reduces that expression to —

du W _
dx + dy' ~ °

which has identically the same form as the equation of continuity
for incompressible flow. The same substitution in the equation
expressing the condition for irrotational flow, whether compressible
or incompressible, viz. —

dv du
vorhcity - - - - = 0,

gives —

and the form remains identically the same.
The circulation round the aerofoil —

K = (udx
Jc

vdy)

VIC] THIN AEROFOILS AT HIGH SPEEDS 261

remains unaltered by the substitution since v and y are changed in
reciprocal ratios whilst u and x remain unchanged . Hence the lift of the
aerofoil per unit of its length, being equal to pQUK, remains unaltered.

This mathematical analogy between compressible and incom-
pressible flow at the same undisturbed speed past a thin aerofoil
having the same circulation implies an important difference in
order that the boundary condition may be satisfied in both cases.
Since v/u in the compressible flow is everywhere less than v'fu, and
the flow adjacent to the aerofoil profile must be tangential thereto,
the incidence of the aerofoil section must be reduced in the same
ratio, viz. (1 — Afa)1/2 : 1. Whilst this requirement can evidently be
satisfied with vanishing camber and thickness, appreciable camber
and thickness would usually involve as a secondary effect a change
in the shape of the profile in the neighbourhood of the nose ; but
the analogy breaks down for another reason near the nose, viz.
that neither v nor u — U can be regarded as small in this region.
The analogy also requires an expansion in the jy-direction of the
linear scale appropriate to the incompressible flow, and this is
compatible with the reduction of incidence of the aerofoil because
points on the profile, or on any other streamline, in the incom-
pressible flow case do not correspond to points on a streamline in
the compressible flow case.

Thus the effect of compressibility, consistently with the present
approximation, is to enable a thin aerofoil to generate a given lift
at an incidence reduced in the ratio
(1 — Afa)1/3 : 1 compared with the inci-
dence required with incompressible flow
of the same speed. It follows that the
lift-curve slope, dCL/d<x,t is increased by
compressibility in the ratio 1 : (1 — M a)1/3. dc
This result is usually stated as an increase "Ha
in the same ratio of the lift coefficient
for a given incidence.

I40Q. Comparison with experiment. — FIG. IOOE.

Fig. IOOE relates to some well-known

experiments * at high speeds carried out at the National Physical
Laboratory on an aerofoil having the section inset. The obser-
vations are shown as encircled points, while the increase of
dCL/da, according to the above theory is indicated by the full-line.
Close agreement is seen between M = 0-25 and 0-5. Between 0-5
and 0-7 the slope of the experimental lift-curve still increased

* Stanton, A.R.C.R. & M., No. 1130, 1928.

262 AERODYNAMICS [CH.

notably, but at less than the predicted rate. At some undetermined
value of M in the neighbourhood of 0-7 the lift-curve slope began to
decrease, as indicated schematically by the dotted extension to the
experimental curve, to a much reduced value at M = 1-7.

The section of the above aerofoil may be regarded as favourable
to the conditions postulated in the theory except that the thickness
ratio was necessarily too large (0*1). Another aerofoil, of more
normal section, showed a less increase of dCL/d<x, and an earlier
maximum ; others have shown in more recent tests * a substantially
greater rate of increase of lift-curve slope than the theory predicts.
Thus experiments so far published suggest that the theory provides
a fair indication, but no more, of the effects of compressibility on
the lift of aerofoils up to moderate Mach numbers.

The value of M at which dCL/d& changes sign is called the critical
Mach number for the aerofoil and the phenomenon is known as the
shock stall (Article 66C). It marks the formation of a shock wave,
attached to the aerofoil at or near the position of maximum velocity
round the profile. The critical Mach number is
sometimes described as that at which this maxi-
mum velocity attains to the local velocity of
sound. However, it has recently been
questioned f whether a shock wave necessarily
forms at this stage. In any case there appears
no reason for supposing that the shock stall
must occur at M = 0-6-0'7, as so often observed,
but rather that aerofoil sections can be designed
to delay this stall appreciably.

I4OR. Supersonic Speeds — The Mach Angle

When a body moves through air at a velocity
greater than that of sound a shock wave pre-
cedes it in the form of a bow wave, in order to
divide the air and deflect it round the nose.
Such bow waves are familiar in photographs of
fast-moving bullets, which show that the dis-
FIG. IOOF. — SHOCK turbance is confined to a thin sheet, Fig. 100F
BY (cf. also Article 66C). Within the sheet,
pressure, density, and velocity change with
very great rapidity. Imagining the air to flow through a stationary
shock wave, its velocity is suddenly decreased and its density

* E.g. Stack, Lindsey and Littell, N.A.C.A.T.R., No. 040, 1938.
t K&rman, loc. cit., p. 263.

VIC] THIN AEROFOILS AT HIGH SPEEDS 263

increased, part of its mechanical energy is converted into heat
and it acquires vorticity. Bernoulli's equation can only be
employed in these circumstances by introducing suitable changes
in the constant, as illustrated in the case of the pitot tube, Article
66D. At some distance from the body, however, the disturbance
becomes small and is propagated at the velocity of sound, whilst air
passing through can satisfy Bernoulli's equation. But there is no
preparatory formation of streamlines ahead such as characterises
subsonic flow, for the body continually overtakes the wave it
generates except for a central region of percussion. Behind the
central region an additional wake is formed, and the streamlines
are parallel to the surface.

Investigation of the complete problem is somewhat complicated,
but progress can readily be made in the case of a thin aerofoil at
small incidence by assuming that the disturbance consists only of
a pressure wave, propagating at the speed of sound, and effecting
only small changes.

It is then easily seen that the waves are inclined to the flight path
at a definite angle. Let P, Fig. 100G, be any point on a body

FIG. lOOo. — THE MACH ANGLE.

moving steadily in the direction PP' at a supersonic speed C7, and
let it reach the position P' at the end of an interval of time t, so
that PP' = Ut. A small disturbance of pressure starting from P
will in the same interval of time travel a distance at, a being the
velocity of sound, so that in the two-dimensional case the wave
front will lie on a circular cylinder of radius at whose axis passes
through P. Similarly, half-way through the interval of time when
P has reached a point P" such that PP" = \Utt the wave front will
lie on a circular cylinder of radius \at and axis at P", and so on.
Thus no disturbance can have been propagated during the time t
beyond the pair of planes through P' tangential to all such circular
cylinders. Each of these planes is inclined to the flight path at
the angle sin""1 (a/U), which is known as the Mach angle and denoted

264 AERODYNAMICS [CH.

by m. The above argument is readily modified to apply to three
dimensions, the wave front then becoming a cone of angle 2m at
the vertex. The wave fronts are propagated normally to them-
selves, i.e. obliquely through the oncoming stream.

It must be observed that the foregoing result depends upon the
loss of velocity normal to the wave suffered by the oncoming
stream being small. In the case of a large disturbance the velocity
of propagation may greatly exceed that of sound, so that the wave
front is much more steeply inclined. The latter condition may be
expected in the immediate vicinity of a fast-moving body, but since
a large disturbance tends to die away as the wave proceeds, the
Mach angle will still characterise the outer parts of the wave.

1408. Ackeret's Theory

The simplifying assumption above mentioned, viz. that for thin
aerofoils such as those examined in Article 140P the disturbance
may be regarded as everywhere small and the aerofoil flow as nearly
uniform, was introduced by Ackeret * in advancing the following
approximate method of calculating the lift, drag, and pitching
moment at supersonic speeds.

Let the relative velocity U now exceed the velocity of sound aQ
in the undisturbed fluid, so that M > 1. The velocity potential
is related in the same way to the relative motion as for incom-
pressible flow, and substitution in (iii) of Article 140P gives, on
writing na for M2 — 1,

^-tf^O

dy* dx*

This equation has the general solution —

<£ = fi(* - ny) + MX + ny\

The solution over the upper surface of the aerofoil may be regarded
as that for a uniform flow plus a function of the type/x, whence it is
seen that the increment of <f> to be added to that for the uniform
flow is constant along the straight lines y == %/n + constant.
These lines are inclined to the direction of motion of the aerofoil
at the angle —

i.e. at the Mach angle m.

The wave under the aerofoil is similarly treated, leading to
Fig. 100H, where lines have been drawn at the Mach angle from the
nose and tail of the aerofoil. Each pair of lines contains between

* Z.F.M., vol. 16, 1925.

VIC] THIN AEROFOILS AT HIGH SPEEDS 265

them a sound-wave propagating obliquely upward from the upper
surface and downward from the lower surface.

The increment of <f> additional to that for a superposed uniform
flow is constant along the wave front and along any line parallel
thereto within the wave. Hence
the additional velocity u is con-
stant along, and directed nor-
mally to, all such lines. The
magnitude of u appropriate to
any such line is determined by
the boundary condition and so
depends upon the shape of the

aerofoil profile. Considering an / / \ "lie

element of the profile inclined,
as in the figure, at a small angle
z to the relative motion, the FIG. IOOH.

component of fluid velocity

along the normal to the element is u cos (m — e) and the
component of the velocity of the element itself in the same direction
is t/e. These must be equal, whence —

u = Us sec m, . . . (ii)
provided e is small.

Assuming now the air to be flowing past the stationary aerofoil,
the pressure p within the standing sound-wave is related to the
undisturbed pressure p0 by (52) 'of Article 31, since increase of
p is small ; and p — pQ may be expanded as described in that
article since the resultant fluid velocity q within the wave differs
little from [7, giving approximately —

— u

Now q has the components U — u sin m and u cos m, and sub-
stituting,

p - pQ = Ipoff/1 — [([/- u sin mY + (u cos m)9]}.
Since u is small, terms involving its square may be neglected, giving —

p — p0 = p0C7w sin m.
Substituting for u in this equation from (ii),

p — p0 = p0C72e tan m.
But by (i) tan m = aQl(U* - a02)l/2. Hence finally—

• L rk _

. . (iii)

JPot/« (M» - I)1'2' '
where M , the Mach number, = Ufa0, as before.
A.D.— 9*

266 AERODYNAMICS [CH. VI C

The pressure coefficient, given by (iii), may be integrated round
the profile of a given aerofoil section, in the manner described in
Article 44, to yield estimations of the lift, drag, and pitching-moment
coefficients, ignoring skin friction. For M = 1-7, Taylor * examined
from this point of view the biconvex circular-arc aerofoil of Fig, 100E,
and obtained good agreement with Stanton's experiments. Approxi-
mately, the calculated value of dCJdat. is 2-85 and the observed
value 3. Comparison with the experimental value of 4*85 for
M = 0-5 illustrates the loss caused by the compressibility stall.
At an incidence of 7 \Q the drag coefficeint (CD) was observed to be

nearly 0-1 and the calculated value,
neglecting skin friction, is about 7
per cent. less. Few data have yet
been published regarding tests on
aerofoils of other sections at super-
sonic speeds.

I40T. Like Glauert's theory for
thin aerofoils at subsonic speeds,
Ackeret's theory for the higher range
is regarded as an interesting and
simple approach to a difficult matter,
achieving success in favourable cir-
cumstances, and likely to be improved
or adapted as more experience is
gained in this comparatively new but
important branch of our subject.

It is clear that the assumption of
sound-waves cannot be justified in
the region of the nose of the aerofoil
at a very high speed, and that the
shock wave there formed will involve
considerations of the kind investi-
gated for the pitot tube, Article 66D.
More accurate methods due to Prandtl and Busemann are also
available for determining the flow over the profile. The matter
is developed further in a paper by Hooker, f from which Fig. 1001 has
been prepared with reference to the above biconvex aerofoil and Mach
number. Considering that skin friction is neglected, the agreement
between prediction and experiment is seen to be good. The
maximum experimental L/D is only 3^, but would be greater for a
thinner section.

* A.R.C.R. & M. No. 1467, 1932, f A.R.CR & M. No. 1721, 1036.

-0-1

° 4° 6°
INCIDENCE

Theory (Hooker)

O O Experiment (Stanton)

FIG. lOOi. — BICONVEX AEROFOIL
AT M = 1-7.

Chapter VII
VORTICES AND THEIR RELATION TO DRAG AND LIFT

GENERAL THEOREMS AND FORMULAE

141. In Chapters V and VI the motion of the fluid was assumed to
be wholly irrotational, but experiment shows motions of practical
interest to comprise rotational and irrotational parts (Chapters II
and VI B). While again assuming in viscid incompressible flow, we
now extend its nature to this composite structure. In general the
fluid motions will be unsteady but Bernoulli's equation will apply to
irrotational regions, though not where vorticity exists ; through the
latter there will be a variation of pitot head.

It will be proved that elements of fluid possessing vorticity are
axially continuous with elements similarly characterised ; vorticity
at a point in a cross-section of a bulk of fluid implies the existence of
a string of rotating elements, cutting the section at the point.
Such a string cannot terminate in the fluid, we shall find, but must
either be re-entrant, forming a ring or loop, or else abut on a boun-
dary. The line (in general curved) to which the axis of rotation of
every element of the string is tangential is called a vortex line. If
vortex lines be drawn through every point of the periphery of a very
small area, they form a vortex tube, and the fluid, of which the small
area is a cross-section, is called a vortex filament, or simply a vortex.

A difficulty is sometimes experienced at the outset with the fore-
going definitions, because the tangible evidence of a real vortex in a
wind is usually a widespread swirl of air. But in theory, as in fact,
every vortex has inseparably associated with it an external motion ;
for an inviscid fluid this is an irrotational circulation, a condition to
which air flow approximates under aeronautical conditions.

We begin by considering in detail a simple type of theoretical
vortex which long, straight parts of practical vortex loops resemble,
known as Rankine's vortex.

142. Isolated Rectilinear Vortex of Circular Section and Uniform

Vorticity

The vortex is assumed to be straight and infinitely long. If a is its
radius and £ its uniform vorticity, we have from Article 39 that co, its

267

268 AERODYNAMICS [CH.

angular velocity, = J£. The circulation K round its periphery is
2na . 00. Hence :

K = 2w . 7raa = & . . . (178)

writing a for the area of cross-section. Any of these quantities
defines the strength of the vortex, and this definition is carried over
to vortices of cross-sectional area cr which are not straight.

Outside the vortex the flow is irrotational, and the velocity is
assumed to be continuous at r = a. Therefore, an irrotational
circulation of strength K must surround the vortex. If q is the
velocity at any radius r, we have —

for r < at q = wr = — ~r ;

for r > a, q = — ..... (179)

&TW

Let P be the pressure at r = oo when q = 0. Applying Ber-
noulli's equation through the outer flow gives for r > a —

'->-£• • • «

as was shown in Article 103 to be consistent with the element being in
equilibrium under the pressure gradient and the centrifugal force.
Within the vortex there is, of course, the same condition for equi-
librium, but Bernoulli's equation does not apply. Since we have
assumed constant angular velocity, however, for r < a —

Tr
Integrating and substituting for o> from (178) —

pK*
P = *&** + «»*•

Now this must give the same pressure as (i) when r = a. Therefore
the constant = P — pK*/4n*a2. Hence, within the vortex —

• • • (I80)

Fig. 101 shows the variation of velocity and pressure through the
vortex of the diameter shown. A practical vortex of sufficient
size to investigate experimentally differs in that its spin is not
constant and its periphery is less sharply defined.

The foregoing supplements Article 103, showing the simplest
condition under which irrotational circulation can occur round a

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT

fluid core. To prevent cavi-
tation, P must exceed the
pressure drop at the centre,
which amounts to pJta/47rta1.
For the outer flow we
have, from Article 103 —

The stream function for the
inner flow is obtained at
once as —

269

£ „ FIG. 101.— DISTRIBUTION OF VELOCITY AND

the negative Sign following PRESSURE THROUGH A RANKINE VORTEX.

from choice of counter-
clockwise sense for K positive. For (iii) to agree with (ii) when
r — a, the constant = K/4n and (iii) becomes —

143. A slight generalisation of the above has an experimental
interest. In this the vortex is assumed to be vertical, and in a bulk
of liquid on whose free surface it terminates.

To take account of the weight of the liquid, of density plf equa-
tions (i) and (180) of the preceding article become (cf. Article 6) —

P - P K* f

r — — gz> for r > at

for

Lri = J® d- r*}-

Pl 47U208 \ 2a*J

where z is the depth below the general level of the surface.

Now, over the free liquid surface the pressure must be constant,
and hence a dimple is formed. If z' denotes the depression of the
surface through the dimple—

zf == - — r-.» for r > a,

— — L for r < a.

The maximum depth of the dimple is K*/4gn*a*

270 AERODYNAMICS [CH.

By observing dimples on the surface of water contained in a tank,
the positions of the ends of vortices within the tank terminating on
the surface can be found with some accuracy, although it is usually
more convenient to sprinkle aluminium dust on the surface, which
reveals the streamlines and facilitates photographs. If the vortices
are in the atmosphere above the tank and terminate on the water
surface, liquid is pressed a short distance up into their cores. When
they form loops within an air stream, one way of making them
visible is by introducing water-vapour, which tends to condense in
the interior of the filaments.

144. An essential difference between a complete vortex and
circulation around a solid core is that the latter may be fixed or
constrained to a certain path, whilst the vortex is free to move. The
outer irrotational flow associated with a vortex is called its velocity
field, and the velocity at any point the induced velocity. The
velocity at the centre of an isolated rectilinear vortex in an infinite
expanse of fluid, which is stationary at a large distance from the
vortex, or within a concentric cylindrical boundary, to take another
example, is zero, and the vortex remains stationary. But this is not
the case with a vortex ring or loop, or when one rectilinear vortex is
near another or approaches a boundary.

Although requiring a knowledge of the strengths and instantaneous
disposition of the vortices, Aerodynamical calculations are chiefly
concerned with the velocity field, and it is nearly always permissible
to neglect the effects of a vortex diameter and of the particular
distribution of vorticity within a given vortex. In the following
articles the vortex filaments are assumed to be thin and of uniform
vorticity throughout any cross-section.

We proceed to prove a number of theorems. These are rigidly
true only for the inviscid fluid assumed, but their direct application
to air flow is remarkably fruitful in practical results. The theory of
inviscid vortices was, in the first place, due to Helmholtz, although
further developed by Kelvin.

145. The Strength of a Vortex is Constant throughout its Length

Fig. 102 shows part of a vortex filament, the circuit ABCDD'C'
B'A'A being drawn on its surface. The lines AA', DD' are adjacent,
so that ABCD and A'B'C'D' enclose two sections of the vortex.
Let the circulation round the first section be K and that round the
second K'.

It is clear that K, K' are the same as if evaluated round normal

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 271

cross-sections in the positions A, A'

along the vortex, because, if a denote

the normal cross-sectional area and

£ the vorticity at A, K = £cr, by

(178), which applies to a curved as well

as to a straight vortex, and if the

actual section be inclined at a small

angle to the normal, the component

of spin will be reduced below £ by the

same factor as that by which the area

will be increased above cr, so that the

product £cr remains independent of the angle. Hence K, K' give

the strengths of the vortex at A, A'.

Now, the cylindrical surface having ABCDD'C'B'A'A as boundary
may be split up into a large number of elements of area by a fine
network of lines drawn in the surface, as was done in Article 97, and
as in that article the circulation round the boundary will equal the
sum of the circulations round the elements enclosed. But the sur-
face does not penetrate into the vortex, and consequently the
circulations round its elements are all separately zero. Hence the
circulation round the boundary is zero. Also, the flow along AA'
cancels the flow along D'D. Therefore, the circulation round
ABCD equals that round A'B'C'D', i.e.—

K = K' or fr = £ V.

If the angular velocity within a vortex filament varies along its
length, the cross-section also varies, or vice versa, in such a way
that the product of the angular velocity and the cross-section remains
constant.

146. Other Vortex Laws

A theorem given in Article 97 can now be re-stated as follows :
The circulation round any circuit is equal to the sum of the strengths
of the vortices it encloses. It should also be noted that this
theorem is not restricted to two dimensions.

Let a wide circuit move at every point with the fluid and enclose
a single isolated vortex section. Then by the above and Kelvin's
Theorem (Article 140K) the strength of the vortex is constant with
respect to time. Since also the strength of a vortex is constant along
its length, a vortex cannot come to an end in a perfect fluid, but must
either be re-entrant (like a smoke-ring) or abut on a boundary (as
described, for instance, in Article 143).

272 AERODYNAMICS [CH.

Now reduce the circuit to a loop which at some instant encircles
the isolated vortex section closely, and let the loop subsequently
move with the fluid so that the circulation round it remains constant.
But this circulation is equal to the strength of the vortex originally
enclosed, itself constant. Hence the vortex remains enclosed ; i.e.
a vortex moves with the fluid.

The above laws are of such outstanding importance in Aero-
dynamics that brief comment on their modification for a viscous
fluid such as air may be interpolated here. In ordinary circumstances
deviation from Kelvin's Theorem is negligible. On the other hand,
concentrations of vorticity diffuse outward, like heat. Thus a real
vortex tends to remain of constant strength but to increase in
diameter. The laws for a perfect fluid are effectual in air away
from boundaries, including that a vortex cannot originate or termin-
ate within the fluid. Vortices may be built up slowly but originate
from the surfaces of moving bodies by the action of viscosity in the
presence of intense velocity gradients.

147. It is seen that the motion of a vortex arises, not from itself,
but from the general field of flow, which may be due to a number of
causes, such as sources, sinks, and other vortices. A vortex line
very close to the surface of a body in motion through air actually
also moves with the fluid, because of the real boundary condition of
absence of slip and the action of viscosity in making the velocity of
the fluid adjacent to the vortex line almost equal to that of the body.
It will occasionally be convenient to treat of a vortex line constrained
to move with a body, while ignoring viscosity and the real boundary
condition. The vortex line is then said to be bound.

148. Formulae for Induced Velocity of Short Straight Vortices

The derivation of formulae for the velocity components at a point
due to one or more vortex loops is beyond the scope of this book.
It is shown in Hydrodynamics that each element of fluid possessing
vorticity implies an associated increment of velocity in every other
element of the fluid mass. The direction of this velocity increment
at any point is perpendicular to the plane which contains the point
and the axis of rotation of the vortex element. If 8q denote the
increment at P due to a small length, 8s of a curved vortex filament of
strength K distant r from P, and 0 the angle between r and 8s, it is
found that —

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 273

The total velocity q at P due to the whole filament is obtained by
integrating along the filament.

In the case of a straight finite length QR of a vortex, the velocity
at a point P distant h from its axis (see Fig. 103) is perpendicular to

the plane PQR and from (181) amounts to :

_ # fRsinjh ds __ Kh CRds
4?u J Q r* 4:71 J Q r*'

Changing the variable from 5 to y, since ds/dy = h sec1 y

sec2

where the limits are now from — ( r —

K .

cos a)*

to p. Hence —

. (182)

An application of this formula occurring frequently in aerofoil
theory is when P is at a distance x, measured along the vortex, from
one end of a straight vortex length, whose other end is a long distance
away. We then have —

l-All + 77/IiVlsl ' ' ' <183)

(184)

i.e. one-half the induced velocity for a rectilinear vortex.

149. We now consider some two-dimensional vortex fields and
vortex motions of importance as leading, to approximations to those

If P is opposite the end of a semi-infinite straight length,

274

AERODYNAMICS

[CH.

occurring in aerofoil theory. For ease of future reference, the vortex
filaments are assumed to extend indefinitely in both directions

parallel to the axis Ox.
In the jtf-plane perpen-
dicular to them v is the
velocity component in
the direction Oy> w
that in the direction
Oz, and the third com-
ponent u = 0.

Vortex Pair

The combination of
two parallel rectilinear
vortices of equal and
opposite strengths is
called a vortex pair.
Let them be situated
instantaneously on the
jy-axis at A and B
(Fig. 104), equidistant
from the origin and
distant / apart, and let
their strengths be ^ K
as shown. Neither has

any motion due to itself, but each has a velocity induced by the

other, given by —

FIG. 104.— INSTANTANEOUS STREAMLINES OF A
VORTEX PAIR.

Below : Construction for the resultant induced
velocity.

(185)

Thus the vortices move in the direction Oz, i.e. downward in the
figure, at this constant velocity, remaining a constant distance apart.
Two sets of streamlines arise, viz. those relative to the fixed axes
of reference and those relative to the vortices. The first are identical
with the equipotentials of a source and sink occupying the positions
of the vortices, and might be inferred from Article 104. But
directly, since for A and B alone —

= _ „ log

respectively, we have for the combination —

. (186)

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 275

The streamlines are shown in Fig. 104, but represent only an
instantaneous plotting, for A and B immediately move away from
the ^/-axis. In accordance with Article 21 they are more appropri-
ately called path-lines.

To obtain the steady streamlines relative to the vortex pair, we
add to the field of flow a velocity — w't or to fy the increment
— w'y = — Kyfiid, obtaining —

[°g?+f)' ' ' (187)

These streamlines are shown in Fig. 105. It will be noticed that

fluid particles contained within a

certain oval accompany the vortex

pair in its career. The streamlines

external to the oval represent the

flow past a cylinder of this section

broadside on to the stream. The

dimensions of the oval are 1-05/ by

0-87/, approximately.

The instantaneous velocity rela-
tive to the fixed axes of reference
at any point P (yt z) in the field is
readily obtained by the construc-
tion shown in Fig. 104. Careful
note should be made that the ve-
locity of the vortices does not
affect this velocity, just as the speed
of a star makes no difference to the speed of the light it emits.

The velocity is equally simply found in analytical terms. For
example, the component w at P is given by —

V-y

FIG 105.— STEADY STREAMLINES
RELATIVE TO A VORTEX PAIR.

W =

I f ^-

1/17 ..\»

(188)

Along the y-axis w represents the true instantaneous velocity, and
the formula for it is —

w =

. ,

(189)

V-y\

The distribution of w along Oy is shown in Fig. 106, where the vortices
have been given an appreciable size, but subsidiary effects of this
size have been neglected. When P lies between A and B, w is posi-
tive, i.e. downwardly directed in the figure ; midway between them
the velocity is four times as great as the velocity of the vortices.

276

AERODYNAMICS

FIG. 106. — DISTRIBUTION OF VELOCITY
THROUGH A VORTEX PAIR.

[CH.

Beyond A or B, w is nega-
tive. The special interest
of this example in connec-
tion with aeroplane wings
will be described later.

150. If two vortices A,
B distant / apart have
strengths Klt Kz, both of
the same sign, the velocity
of A will be K.z/2nl, while
that of B will be K^nl.
Both velocities will be per-
pendicular to the line AB,
but in opposite directions.
Thus the line AB has a steady angular velocity and one point on
it, which is easily found, remains fixed if a velocity is not induced
there by other causes.

The instantaneous velocity at any point due to any number of
parallel vortices may be found by the superposition of the velocities
induced by the several filaments. The fixed point in the last example
corresponds to the C.G. of the two vortices if each is imagined to be a
gravitating line of mass equal to its strength. The analogy can be
applied to more complicated dispositions of vortices, and so an axis,
parallel to Ox, can be found for the system which remains fixed (in
the absence of other disturbances) as the vortices move. But in the
important case when the algebraic sum of the strengths vanishes,
i.e. for a group of vortex pairs, the axis is at infinity.

There exists another analogy, of considerable experimental use,
viz. that between a vortex filament and a wire conducting an electric
current. The lines of magnetic force surrounding a current, or group
of currents, which can readily be mapped out by experimental means,
correspond exactly to the streamlines of the vortex case. The
analogy finds practical expression in an apparatus called the
' electric tank.' *

EFFECT OF WALLS
151. Applications of the Method of Images

When a vortex approaches a parallel boundary which is not co-
axial with it, the streamlines become distorted, and a motion is

* Relf, A.R.C.R. & M., 905, 1924.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 277

induced in the vortex. These effects may sometimes be determined
by the method of images. The question is one of outstanding
interest, because Aerodynamical measurements of flow involving
vortices are usually made in the presence of walls.

As a first example, let a single rectilinear vortex of strength K be
distant \l from a plane rigid wall parallel to its axis. The presence
of a rigid boundary requires that at every point on it the normal
component of velocity shall vanish. In the present case this is
obviously satisfied by imagining a second vortex of strength — K to
be situated at a distance %l beyond the boundary, opposite to the
real vortex, i.e. by introducing the image of the vortex in the wall.
The streamlines are determined by the two. The flow round, and
the motion of, the real vortex will be exactly those that would
obtain if it formed one member of a vortex pair. The solution is
otherwise evident, for clearly the plane xOz in Article 149 might have
been made rigid without effect.

152. Another example, which leads to an important case, is
provided by a rectilinear vortex eccentrically situated within a long
tunnel of circular section. The
boundary takes the place of
one of the co-axial circles of
Fig. 104, which form the in-
stantaneous streamlines of a / / / "A\ ^\»^^ B
vortex pair. The effects of the
tunnel are reproduced by in-
troducing the image of the
vortex in the circular wall, an
imaginary vortex of strength
— K situated at the inverse
point. FIG. 107.

Let A, Fig. 107, be the real

and B the imaginary vortex, and P any point on the boundary.
The condition that the radial velocity component shall vanish at
P whatever its position is that the locus of P be a streamline, and
for this, rA/rB — const.

Bisect the L BPA internally and externally by PC, PD ; BA is
then divided internally at C and externally at D, and —

EC BD rB

— = y-r = — = const.

CA DA rA

Thus C and D are fixed points and, since the L CPD = re/2, P
traces a circle on CD as diameter. Let the radius of this circle

278 AERODYNAMICS [CH.

be a and its centre 0. Then from the above equation and the
figure —

OB =

a =

DA .BC
CA a

(a + r) (OB - a)
a — r

giving—

(190)

Thus a vortex of strength K distant r from the centre of a tunnel
of radius a moves in a concentric circular path with the constant
velocity —

_ K J_ __ ___ # _r_

~~ 2w ' aa/y — r ~~ 2?c " fl1 — r2"

The instantaneous streamlines within the tunnel follow from

Fig. 104.

The image system
for a vortex pair, each
of whose members is
distant r from the
centre 0 of the tunnel
is shown in Fig. 108,
each of the two images
being situated on the
radial plane contain-
ing the corresponding

FIG. 108.— IMAGE SYSTEM FOR A VORTEX PAIR real vortex and dis-
IN A CIRCULAR TUNNEL. tant ^ frQm Q

Each vortex describes

in general a D-shaped path in its half of the tunnel section. But
chief interest attaches to the effect of the tunnel on the velocity
field. In, for example, the particular case where the vortices lie on a
diameter, if their distance apart is then /, w at 0 is given by —

w =

2n ( I a*
Without a boundary we should have at 0

K /4

• (191)

Thus the tunnel wall considerably reduces this velocity ; when
/ as at for example, the decrease amounts to one-quarter.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 279

153. Some further examples of image systems will be referred to
briefly.

FIG. 109.

IMAGE SYSTEM FOR A RECTILINEAR VORTEX BETWEEN PARALLEL WALLS
(THE Row OF IMAGES EXTENDS TO INFINITY).

(1) A vortex parallel to two parallel plane walls gives rise to a
doubly infinite row of images (Fig. 109). If h is the distance apart
of the walls and z the distance

of the vortex from one of them,
the vortices are separated al-
ternately by the distances 2z
and2(A — z).

(2) A vortex in the corner
between two plane walls which
meet at right angles calls for
three images situated at the
corners of a rectangle, as shown
in Fig. 110.

FIG. 110. — IMAGE SYSTEM FOR A RECTI-

'" A RlGHT-ANGLED

(3) The important case of a
vortex pair contained within a

rectangular tunnel leads to a complicated arrangement of images
in doubly infinite columns and rows, as shown in Fig. 111.

Reference will be made
later to this case, when
the subject will be sys-
tematised in connection
with Wind-tunnel In-
terference. But when
the sides of the rectangle
are equal, or nearly so,
it is often sufficient to
substitute for the actual
boundary an approxi-

FIG. 111. — IMAGE SYSTEM FOR A VORTEX PAIR mation consisting of a

IN A RECTANGULAR WIND TUNNEL OF EN- rirrl^ drawn fairltr

CLOSED TYPE (THE COLUMNS AND Rows OF CJrC1C . Of*^ lairlY

IMAGES EXTEND INFINITELY). through the Sides.

280 AERODYNAMICS [CH.

154. Application of Conformal Transformation

It will be apparent that the method of images is convenient in some
cases of vortices near boundaries but cumbersome in others. The
total effect of an infinite series of images may be difficult to sum
while the step-by-step increments may converge so slowly as to make
approximate calculation laborious. In some cases, again, an image
system cannot be found. An alternative method of solution is
provided by the use of conformal transformation which may be
applied to parallel rectilinear vortices.

For ease of reference the vortices are now assumed to be perpendic-
ular to the A;y-plane, i.e. our real plane will be the z-plane of Article
122, where the co-ordinate of a point is z = x + iy, and transforma-
tion will be made to a tf-plane where the co-ordinate is t = £ + ir\.
If at any instant there is a vortex at the point zl in the real plane, at
the same instant a vortex of equal strength will exist at the corres-
ponding point ti in the transformed plane. There are several ways
of proving this. The most direct is to note that outside the vortex
a velocity potential <f> exists ; that the strength is equal to the
interval in <£ once round a circuit embracing the vortex ; that <j> has
the same value at corresponding points in the two planes ; and so, if
the interval of <f> be evaluated round corresponding circuits in the two
planes, it will clearly come to the same thing. But it will be seen
that, the transformation having changed the boundaries and the
geometrical dispositions of other vortices that may be present, the
vortex particularly considered will not move along a path in the t-
plane which corresponds to its path in the z-plane. The two paths
can be related, so that one can be drawn from the other, but, since in
Aerodynamics we are chiefly concerned with instantaneous induced
velocities, this development will be left to subsequent reading.*

The general aim in applying conformal transformation is to
simplify the configuration of the real system, so that a convenient
arrangement of images can be used ; the difficulty, as in problems of
potential flow, lies in finding the transformation formula. The
simplest image system is that appropriate to a single vortex near a
parallel plane wall. If in the 2-plane the trace of the wall coincides
with the #-axis and the real vortex of strength K is at the point
xl9 3/1, the image of strength — K will be at the point xl} — ylt and
the stream function of the velocity field is obtained from (186) as —

* Routh. Proc. Lond. Math. Soc.. t. XII, 1881.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 281

K (x-xp + b-yp

47r g (x - x,Y + (y+ yj*

In the same way the identical system in the /-plane (not that ob-
tained by transformation) gives —

* - - * W lini'-^HlH^8 (192)

*- log * ' • (192)

the real vortex being situated at the point ^, v^, and the wall co-
inciding with the £-axis. If a transformation formula can be found
to convert whatever configuration exists in the z-plane to this con-
figuration, for instance, in the /-plane, the problem in the z-plane is
at once solved.

J55* Vortex Midway between Parallel Walls

The image system for this case follows from Article 153 (I), but
we shall ignore this and solve the problem by the method of the
preceding article. Assume that the distance apart of the walls is
H. Choose Ox in the £-plane, so that y = 0, y = H represent the
walls and Oy passes through the vortex. Let %' = x/H, y' ~ y/H.

Consider the transformation formula —

log t = Ttz/H .... (193)
or —

= c** (cos Try' + i sin Try').
Separating real and imaginary parts —

£ __ gKX' CQS ^yt ^ ^ __ £!* g^ ^y^ ^ ^ ^

Corresponding toy =0, we have £ = e"*', 73 = 0, and corresponding
toy — H we have £, = — en* ', YJ = 0. Thus both walls in the z-plane
transform to the £-axis in the /-plane. The vortex of strength K at
%' = 0, y' = ^ in the 2:-plane transforms, using (i), to a vortex of
strength K in the /-plane at the point J; = 0, 73 = 1.

Thus, in the /-plane we have a vortex on the Y)-axis at unit distance
from a single wall coinciding with the £-axis. Therefore, the stream
function of the velocity field in this plane is given by (192), and
comes to —

f K i™ P + to - l)'

+ = -s^5TRn--ir'-

To find the stream function in the 2-plane, substitute for i;, TJ in
terms of x't y' (cf. Article 122) from (i) obtaining —

282 AERODYNAMICS [CH.

_ ^ #** cos* re/ + (e** sin ity' — 1)»
* ~~ ~ 4w °g e2™' cosa Try' + (^ sin Try' + 1)'
K f d2"*' — 2071*' sin Tty' + 1

— Jog _ £

4?r 6 *** + 24"* sin Try' + 1

This expression reduces on dividing the numerator and denominator
of the logarithm by 2e**' to —

K cosh nx/H — sin ny/H

* = " 4^ °g cosh 7rA;/ff + sin ny/H' ' ' ( '
The path-lines are shown in Fig. 112.

FIG. 112. — STREAMLINES FOR A RECTILINEAR VORTEX MIDWAY BETWEEN
PARALLEL WALLS.

A particular interest centres in the effect of the walls on the
velocity midway between them. This velocity is given by —

K f sinh TixjH . sin icy/H 1
H/2~ 2^ Lcosh2 nx/H — sin2 ny/H J y,_1/2

-^ X (195)

"" 2H • sinh «/J? ...... ( }

In the absence of the walls, the velocity along this line would be
equal to Kftnx. Hence the walls, if at distance H apart, reduce
the velocity midway between them in the ratio —

™ ..... (196)
sinh KX/H v '

At a distance behind the vortex equal to its distance from either wall,
for example, the reduction is over 30 per cent.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 283

155 A. Bound Vortex in Stream between Walls

The necessary modification of (194) to give the stream function
for incompressible and irrotational flow past a bound vortex mid-
way between parallel walls will be evident from the preceding
article. Mention was made in Article 66A of a method of obtaining
the lift of an aerofoil in a wind tunnel by integrating the changes
of static pressure on the floor and roof of the tunnel. Further
investigation in the present article will be directed more particularly
to this question under the conditions stated and assuming the
aerofoil to be small in chord, so that it may be replaced by a simple
vortex.

The velocities along the floor and roof due to the vortex alone
are, from (194),

[3^1 ___ K I" cosh rex' cos ny'

3yJ^o,i ~~ "

Adding a uniform stream of velocity — U to produce an upward
lift L = $KU per unit length of the bound vortex gives for the
resultant velocity along these walls —

?o = — U + UQ and ql = — U + uv . (ii)
where the suffixes 0 and 1 distinguish the floor and roof, respectively.
Now, the flow being irrotational and incompressible, Bernoulli's
equation gives —

-=: ( _—

u \U
K

_____ _ K y

\2Wcosh^r#' ^ V ~~ \2UH coslTTu*' ~ V
by (i) and (ii), or —

UH cosh nx'
L

~ H cosh TT*' ^

Thus if a gauge be connected between two static pressure holes at
x = 0, one in the floor immediately below the lifting vortex and the
other in the roof immediately above it, the lift per unit length will
equal H x the pressure difference recorded. We also verify that —

poo AGO

L = (p0 - pjdx = oKU

J-CO J-a

- pKU.

284

ROOF

x/H

1-0

FIG. 112A. — LIFTING VORTEX
BETWEEN PARALLEL WALLS.

AERODYNAMICS [CH.

A greater fraction of the lift will
clearly be supported by the roof than
by the floor, and the ratio of the two
contributions to the total reaction is
readily determined. The two pressure
distributions are plotted in Fig. 112A.
If the experimental exploration of
PQ — pl extends only to a distance X'
on each side of the small aerofoil, the
fraction of the lift obtained by integra-
tion will be —

dx' 2f ,1*'

, - - tan"1 ?* (iv)

_ nx TU L J_^'

This result is plotted in Fig. 112B.

1558. Other Applications of the Transformation

In applying this method to other problems of flow between
parallel walls it is sometimes advantageous to modify (193) to log
t = 27uz', which may be written, if r, 0
are polar co-ordinates in the /-plane,

log r + i0 = 2(nx' + try'),

giving log r = 2nx', 6 = 27cy'. Thus

r — I when x = 0, i.e. thejy-axis of the

2-plane corresponds to a circle of unit

radius in the /-plane ; also 6 = 0 when

y = 0, and 0 = ± ^ when y = ± \H.

It follows that the whole of the /-plane

yields in the 2-plane an infinite strip

of width Ht as before, but with the

#-axis midway between its edges (Fig.

112c). These edges are derived from the two sides of the negative

half of the real axis in the
/-plane. Hence the strip may be
regarded as the section of a wind
tunnel provided the correspond-
ing flow in the /-plane makes
the negative half of the real
axis a streamline. A uniform
flow between the parallel walls

corresponds to a source at the origin in the /-plane since lines

radiating at equal angle-increments from that origin transform

FIG. 1 12s. — INTEGRATED WALL
PRESSURE.

t -plane

2 -plane

FIG. 112c.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 285

to equally spaced lines parallel to the walls in the 2-plane, and r = 0
corresponds to x = — oo. If the uniform flow has a velocity Ut
the strength of the source is clearly UH.

(1) Source in Uniform Stream. — As a first example, let a source
of strength m be located in the stream at the origin z = 0, which is
not a singular point. The corresponding disturbance in the £-plane
is an equal source at the point t — I.

The potential function in the tf-plane is —

UH. . m

log (* - 1).

Substituting gives for the 2-plane —

so that the complex velocity u — iv between the parallel walls is

dw _ m £**'

~dz = + H ?^=T

When z is large and negative, the velocity = U, as assumed, and
when z is large and positive the velocity = U + m/H.
On the walls, z = % ± iH/2t giving —

u = U +

H eM + 1
= U + iw(l + tanh Tix')jHt

from which and Bernoulli's
equation may be obtained the
pressure distribution along
the walls, illustrated in Fig.
112D. The streamline fy — 0
will differ from that found
in Article 106, becoming
parallel more rapidly,

(2) Doublet in Uniform
Stream. — Let the source at
z = 0 now be replaced by a
doublet of strength — (ir Then
a doublet of strength ^ appears at t = 1, and in that plane —

UH. . . LL

FIG. 112D.

286 AERODYNAMICS [CH.

so that —

and —

dw „ y., e*"'
u — iv = — = U — ~~T.

dz H ' (*" - 1)*'

On separating (i) into real and imaginary parts it is easily found
that the streamline fy = 0 is given by —

~~ 47cC7 ' cosh 27r#' — cos 2TC/

This deformed circular boundary intersects the jy-axis at points to
be obtained from —

*y tan ?y __

" ""

and the #-axis at the points

sinh'f -

Substituting, for illustration, y = \H in (ii) gives ^/U = TC// and,
by (iii), ^ = 0-254/f. Thus the deformation of a circle whose
diameter is so great as one-half the height of the tunnel is small.*

It is assumed in the foregoing that the strength of a source, not
situated at a singular point, remains unchanged on transformation.
Proof follows immediately from that for a vortex (Article 154) on
substituting fy for <£. The same is not true, however, of a doublet
since the strength of a doublet is proportional to the product of
that of a source and an infinitesimal length. Thus whilst q oc l/r
for a source as for a vortex, q oc l/r* for a doublet. It follows that
the strength [xt in the *-plane is equal to p\dt/dz\, as may be proved
in other ways. In the above example,

dt
dz

27C

Investigations of the kind considered in this and the preceding
article become of interest in the estimation of tunnel constraint on
large two-dimensional models in comparatively small streams, and
the use of adjustable walls to compensate (cf. Article 66A).

* Lamb, Hydrodynamics, 6th ed,f p. 72, where the problem is solved by the
method of images.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 287

GENERATION OF VORTICES

156. As soon as an aeroplane, say, starting from rest, attains
appreciable speed, the air flow induced past its various components
becomes of the general nature described in Chapter II. But the
wakes behind some of its parts — e.g. wings, wheel fairings, or thick
exposed struts — are found to contain discrete vortices. The theory
of the preceding articles then has a practical utility, which depends,
however, upon a knowledge of the vortex distribution and strength.
This information rests principally on observation, because it is
difficult to calculate precisely how the vortices are formed, though it
has been seen that they result from viscosity.

By starting a body of simple shape from rest in a tank of water,
and sprinkling the water surface with aluminium dust, a cinema film
can be taken of the accelerated motion. Under similar circumstances
the same sequence of photographs will apply equally to air as fluid,
but they are less easy to secure in air. Such films show vortices in
various stages of growth, as will be described in the following
articles.

It may be stated at once as a general result that photographs
relating to a very early stage of motion accelerated from rest show
path-lines which, even for bluff bodies, approximate closely to those
of potential flow, as might have been anticipated on theoretical
grounds (cf. Articles 98 and 119). The motions finally established
may differ little or considerably from those of an inviscid fluid round
the same shapes, but viscosity requires time in which to bring about
the change.

157. Impulse

We shall have occasion to refer to the impulse of vortex loops.
The external flow associated with an inviscid vortex loop is irrota-
tional, and could be generated instantaneously from rest by an
artificially arranged distribution of impulsive pressure, which can be
calculated. The matter will be illustrated * with reference to the
vortex pair.

Imagine a very long straight elastic membrane of width / im-
mersed in stationary fluid. Let it be acted upon by a distribution
of impulsive pressure, and let it bend transversely in the process in
such a way that its final velocity at every point, attained at the end

* For rigorous mathematical investigation see Lamb's Hydrodynamics.

288 AERODYNAMICS [CH.

of the impulse, is exactly that appropriate to the fluid velocity field of
a vortex pair situated along its edges. Finally, let the membrane
vanish at the end of the impulse. The irrotational motion of a
vortex pair results.

The impulsive pressure was identified in Article 98 with — p^.
Considering any pair of adjacent points, A and 5, on opposite faces
of the membrane, the difference of impulsive pressure between them
is p((£B — <£A), and this again is equal to p/£, if K be the magnitude
of the eventual circulation round lines coincident with the long edges
of the membrane. Now K is constant. Hence, per unit length —

Impulse = ?Kl . . . (197)

More generally it can be shown that the component in any direc-
tion of the impulse which would generate the velocity field of a
vortex loop from rest is equal to —

pXS (198)

where 5 is the projection in that direction of any area that is bounded
by the vortex loop.

158. Vortex Sheets

A vortex sheet is a fluid layer, in general curved, containing a
continuous, though not necessarily uniform, distribution of vorticity.
Its two surfaces, a small but variable distance $n, say, apart, are
formed of vortex lines. Consider a small length 8s of the sheet
perpendicular to the vortex lines. The circulation SK round the
element SnSs = (q — q') 8s, if qt q' denote the local velocities on the
two surfaces of the sheet, for there is no flow along either of
the 8w-sides. Hence, writing 2ca for the vorticity —

2co . 8n = q — q1'. . . . (199)

Since the vortex lines move with the fluid, the sheet will not be
stationary, but will locally have the velocity \(q + <?')•

We have seen (Chapter II) an example of vortex sheet structure in
the boundary layer. The term is more particularly reserved, how-
ever, for a sheet of vorticity out in the fluid, separating two regions
of irrotational flow which have different adjacent velocities. The
thickness 8w may be considered to become indefinitely small while
the product 2to . $n remains finite, when the sheet is formed simply
of a single layer of vortex lines. It is then sometimes called a surface
of discontinuity. In Chapter V the surfaces of bodies immersed in
the stream were surfaces of discontinuity, but in Chapter II we saw
that a boundary layer of small but measurable thickness represents

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 289

experimental conditions. A surface of discontinuity out in the fluid
may be regarded as the ideally simplified, and a vortex sheet of
finite thickness as the practical accompaniment of a sharp lateral
change in velocity. The jump in velocity may be in respect of
magnitude or direction.

159. Production and Disintegration of Vortex Sheets

Three ways may be distinguished in which vortex sheets are
commonly produced.

(1) Considering, for example, a flat plate started from rest into
broadside-on motion, the path-lines at an initial stage closely accord
with those of potential flow (Fig. 80). But their persistence at the
back of the plate calls for very high velocities near the edges, such as
would lead to cavitation there. Thus the flow must break away,
giving rise to surfaces of discontinuity which spring from the edges
and separate flow from the front of the plate from fluid in the wake.
This conception led Helmholtz and Kirchhoff to a theory of drag in
inviscid flow, which, however, we shall not attempt to follow. We
note that vortex sheets must be expected to replace the surfaces of
discontinuity as viscosity makes its presence felt.

The phenomenon is not, of course, confined to the normal plate,
but occurs whenever flow is asked to turn round a sharp edge. For
this reason alone the streamlines of Figs. 81 and 97 could not persist.

(2) In potential flow completely surrounding a body, an element
of fluid passing close to the front stagnation point arrives near to the
back stagnation point with unimpaired energy. If the contour of
the body is convex to the fluid, the element is accelerated during the
first part of its transit by decreasing pressure, and gathers additional
kinetic energy, which is converted without loss into pressure energy
again during the second part of its transit, when it is moving against
a rising pressure. In a real case, the element enters and proceeds
within the boundary layer, and the viscous tractions prevent its
motion from obeying Bernoulli's equation. Kinetic energy gathered
in the first part of its passage soon flags, and the rising pressure of the
second part eventually turns the element back.

Prandtl, in an analogy, has likened the circumstances to those of
a ball rolling in a smooth guide of vertical wave shape, successive
crests being horizontally level. With no frictional resistance of any
kind, the ball, starting from rest at one wave crest, would accumu-
late sufficient kinetic energy in the trough just to reach the next
crest. But the slightest dissipation of mechanical energy would
result in the ball turning back.

A.D.— 10

290 AERODYNAMICS [CH.

Reverting to the fluid motion, a return flow near the rear part of
the surface of a body wedges the boundary layer away. The posi-
tion round the contour where this occurs is called the point of break-
away. It is always found near the shoulder of a circular cylinder,
except perhaps at very high Reynolds numbers, when it may move
back appreciably. On the other hand, in the case of a thick strut
it may be situated at only a comparatively short distance in front
of the trailing edge at ordinarily high Reynolds numbers. The
segregated part of the boundary layer, a film of intense vorticity,

0-25

0.2

FIG. 113. — ISO-VELOCITY LINES FOR THE FLOW PAST AN AEROFOIL AT THE
REYNOLDS NUMBER 2-1 x 10* AND INCIDENCE 9-6°, SHOWING BREAK-AWAY AND

THE VORTEX SHEET.

Linear scale normal to the aerofoil is magnified 8 times.
(Reproduced by permission of the Aeronautical Research Committee.)

becomes a vortex sheet in the fluid. Fig. 113 shows the break-away
from the upper surface of an aerofoil * at a low Reynolds number,
the vortex sheet being easily recognised by the packing together of
the velocity contours. Such a flow would smooth out considerably

* Piercy and Richardson, A.R.C.R. & M., No. 1224 (1928).

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 291

with increase of Reynolds number. The flat lower surface inclined
positively to the stream is free from the phenomenon, the pressure
gradient not reversing here, so that the element, although moving
unequally and doing work in the boundary layer, is never actually
arrested.

(3) Consider a wing that has a lift. Lift can only arise, as
described in Chapter II, from (upon the whole) a greater reduction of
pressure on the upper surface than on the lower surface. Viewed
in plan the streamlines will be inclined inwardly to a greater extent
above the aerofoil than below it. Discontinuity in respect of
direction of flow occurs in a sheet stretching downstream from the
trailing edge, dividing the upper and lower parts of the flow where
they merge behind the wing. Viscosity ensures that this surface
becomes a vortex sheet. The vorticity is zero behind the centre of
span, increasing in strength, but to opposite hand, towards the
edges of the sheet on either side.

A characteristic of all vortex sheets is their essential instability.
By this is meant that the effects of even an infinitely small distur-
bance instead of being damped out in course of time tend, on the
contrary, to develop. In practice, therefore, only a short length of
newly manufactured vortex sheet can ever be found except in
peculiar circumstances. The effect of development of disturbance is
to make the initially thin even spread of vorticity form marked
accumulations ; in other words, the sheet tends to roll up in some
way. As the production of the vortex sheet goes on, so the gather-
ing of vorticity into hoards continues. This crowding together of
vortex lines in patches describes in a qualitative way the formation
of discrete vortices. Their eventual disposition varies greatly, but
for given cases of motion it is usually, although not always, the
same. The following articles examine in some detail two arrange-
ments which are common and important.

1 60. Karman Trail

The most familiar arrangement of vortices is the procession usually
known as the vortex street. Consisting (Fig. 114) of a moving avenue
of evenly spaced staggered vortices, the two rows being of equal
strength but to opposite hand, it characterises the wakes of all long
cylinders of bluff section, and occasionally, at low Reynolds num-
bers, those of streamline cylinders. At first sight it would seem
plausible to expect the vortices to be disposed opposite to one
another, but such an arrangement is unstable. Kirmdn * showed

* Gdtt. Nachrichten, 1911. Cf. also K&rman and Rubach, Phys. Zeits,t 1912.

292

AERODYNAMICS

[CH.

FIG. 1H.-

-PATH-LINES OF THE VORTEX STREET BEHIND A CIRCULAR
CYLINDER.

this theoretically, successfully calculating the layout of the pro-
cession necessary for stability, and also other matters to which
reference will be made. Therefore, the motion is alternatively called
the Kdrmdn trail.

His results state first that if h is the distance between the rows
and / that between successive vortices of the same row —

A// =0-281 ..... (200)

Imagining the body to move at velocity U through fluid at rest at a
distance, the eddy system left behind is not stationary, for each
vortex has a forward velocity induced in it by all the vortices in the
other row. This comparatively small velocity u is the same for all
and, if K is the numerical strength of each vortex, is given by —

u =

(201)
V ;

The frequency ~ of generation of each pair of vortices, one vortex in
each row, is clearly —

~ = V W

The K4rm£n trail may be regarded as the central region of a large
number of very elongated loops, all closed in a zig-zag fashion across
the avenue behind the extremities of the cylinder causing them.
One long vortex length matures during the short time l/~ near to
the surface of the body— behind the shoulder of a circular cylinder,
behind one of the sharp edges of a normal plate, or a little upstream
of the tail of a cylinder of streamline section — while a fully-grown
long vortex is detaching itself from the other side of the cylinder and
beginning to be left behind. If a circuit be drawn round the median
section of the cylinder in such a way as to include the ' bound ' vortex
while excluding the free vortex, conceived as having just been left in
the wake, we find a circulation round the circuit. Therefore, a

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 293

transverse force on the cylinder is expected from Article 109. This
will change in sign periodically, since the next vortex to mature will
be situated on the opposite side of the cylinder. The periodic force
exists in experiment and, in the case of fine wires, leads to singing,
observation of the tone of which provides one method of finding the
frequency of the eddies.

It will be noted that the above equations do not provide a solution
of the problem for any particular Reynolds number. Moreover, the
Reynolds number R as well as the shape of the body affect the trail,
for, if b is the maximum width of section, we easily find, by the
method of Article 47, that —

~=^/(/J) .... (203)

for a given shape. From this we can deduce the variation of ~ for
bodies of different sizes and the same shape at constant /?, but f(R)
remains to be determined experimentally for each shape.

161. Application to the Circular Cylinder

Some observations with a long circular cylinder at a Reynolds
number (= f/6/v, b denoting the diameter) of about 2000 gave
approximately: ~ b/U = 0-2 and «/E7=0-14. From (202)—

~j> _6/ u

~U~~l\ U
and, on substituting the above measurements —

lfb = 4-3.
From (200)—

h = 0-281 / = 1-21 b

showing that the track of the established vortex street was, as is
usual, 20 per cent, wider than the cylinder. Finally, from (201) —

giving the strength of the vortices for a given speed and size of
cylinder.

A first approximation to the drag is obtained as follows : the
mean rate of change of impulse parallel to the direction of motion
required to create the vortex loops from rest at the observed rate is,
from (198)—

= 0-2- x p X 1-7 Ub x 1-21 b

294 AERODYNAMICS [CH.

per unit length, giving a drag coefficient —
~Kh

c°=i™ ......

~b K h

= Ju-ub-b==()'*xl'lxl'21

= 0-82.
Equation (204) may alternatively be arranged in the form —

CD = — ( — ^-" X 2V2 . ul X 0-281 /)

Rigorous examination by Kdrmdn takes account of factors neglected
above, and he shows that the cylinder experiences, still on account of
the vortex street only, a 10 per cent, greater drag coefficient, given
by-

CD- 1-588 -O . . (206)

which yields the value 0-90 from the above measurements.

The drag coefficient obtained by direct weighing at this Reynolds
number is 0-96, approximately. Thus vortex production accounts
for nearly the whole of the drag in the case considered, and investiga-
tion at other Reynolds numbers suggests this to be generally true for
the circular cylinder for R > 100.

Fig. 25 shows the variation of ~ b/U with Ub/v for this shape. At
R = 2 x 10*, frequency begins to increase much more rapidly with
U for constant diameter and fluid. The corresponding decrease in
drag would be consistent with the vortex street becoming narrower
by some 50 per cent.

162. Form Drag

That part of the total drag on a body which can be traced to
the shedding of a vortex street is an important instance of form
drag, which arises from a modification — sometimes a great change —
of the pressure distribution pertaining to irrotational flow, illus-
trated in Figs. 72 and 92B. It is not always correct to ascribe the
whole of the pressure drag integral (cf. Article 44), i.e. the whole
difference between weighed drag and skin friction, to form drag ;
part may be due to a different cause, as will shortly appear, being
associated with the production of lift.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 295

The form drag of long flat plates normal to the stream is nearly 60
per cent, greater than for circular cylinders for R between 10* and
10*. For the finer strut illustrated in Fig. 91, on the other hand, it is
95 per cent, less at R = lo6. Thus form drag, when it is entirely
parasitic in nature, can be reduced greatly by suitable streamlining.
But when required for landing and slow diving it can be obtained
in large measure by exposing a long normal plate (Article 76).

If the body is of short length across the stream — a body of revolu-
tion, for example, such as a sphere or an airship envelope — a vortex
wake may be produced, but it has a different form. We might have
expected the elongated loops of the vortex street to shrink to a
succession of vortex rings, and these are observed at low velocities
behind small spheres in air. But at a greater Aerodynamic scale
the vortices consist of narrow loops in spiral arrangement, the whole
system spinning about a central axis.

Again, breakaway may be prevented by turbulence and a wake
of vorticity formed without discrete vortices.

It is convenient to leave open the question of vortex arrangement
in the wake, and to define form drag as that part which is not due
to skin friction or lift.

APPLICATION TO WINGS
163. Lanchester's Trailing Vortices

We turn now to the important case of Article 159 (3). The lifting
wing of finite span is assumed to be of thin streamline section set at a
small angle of incidence to the undisturbed wind. It is assumed to
have no form drag, the wake arising from its boundary layer being
neglected. The vortex sheet stretching downstream from its trailing
edge, associated with the difference existing in lateral components
of velocity of confluent streams from above and below, splits into
two halves along the centre-line, where the vorticity vanishes, and
each half rolls up about a roughly fore-and-aft axis to form down-
stream one member of a vortex pair. To a first approximation Fig.
106 gives the velocity distribution through a cross-section of the
wake far behind a wing, where the distance between these long eddies
is much less than the span of the wing. They often partly form
close behind the wing-tips, and are then called wing-tip vortices,
but the fully developed motion is a trailing vortex pair. Their
presence was inferred on theoretical grounds by Lanchester in the
course of his pioneering work on Aerodynamic lift.

Remembering that vortex lines cannot terminate or originate away

296 AERODYNAMICS [CH.

from the vicinity of the wing, we are faced with a question as to
what may be the complete configuration of which the vortex pair
forms part. Moreover, it is not clear without further examination
why the wing should exert the lift on which the vortices depend.
These interrelated questions are clarified by following the motion of
an aerofoil from rest, again with the help of photographs of the
formation of the vortex system.

164. Generation of Circulation and Lift

Let the aerofoil, of span 2s, start from a position of rest A in
stationary air and, after a brief period of rapid acceleration, be
moved at constant small incidence in a straight line, assumed for
convenience to be horizontal, at a velocity U of considerable
magnitude. It is assumed that the time to a position H such that
AH is large compared with 2s is sufficiently short for diffusive action
of viscosity on the vortices formed to be neglected ; this is consistent
with the long persistence of vortices observed in a fluid of such small
viscosity as that of air.

During the period of acceleration from rest, the flow closely
approximates at first to potential acyclic motion and is momentarily
of the type illustrated in Fig. 97, the back stagnation line being
situated on the upper surface of the aerofoil well in front of the
trailing edge. The high velocity gradients caused near the trailing
edge give rise to a vortex sheet which begins at once to roll up in
the manner shown at (a) Fig. 115. A pack of vortex lines (b)
parallel to the trailing edge begins to appear near A. This is known
as the starting vortex. Photographs show that as soon as accelera-
tion ends the vortex sheet ceases to be formed and the starting
vortex becomes detached, as at (c) in the figure, and is left perman-
ently behind at the position A.

The foregoing description applies to the vertical plane of sym-

s o

(c)

(b) (a)

Q T R

FIG. 115. — FORMATION OF "THE STARTING VORTEX.

VII]

VORTICES AND THEIR RELATION TO DRAG AND LIFT

297

metry. Let the circuits shown in Fig. 115 be in this plane. The
strength of the starting vortex is measured by the circulation — K
round any circuit SPQTS which encloses the vortex only. Since the
trailing edge of the aerofoil was to the left of ST in the figure on
starting from rest, ST has been cut by the aerofoil. But a circuit
such as OSPQTRO may comprise the same fluid particles. The
circulation round it was originally zero, and still remains so. There-
fore, the circulation round the circuit OSTRO, or any other in the
plane embracing the aerofoil but not the vortex, must be equal to K.
The vortex lines of the starting vortex cannot terminate in the
fluid. We might perhaps conceive of their turning at each end and
abutting on the aerofoil, although this would be difficult to imagine.
The foregoing proves, however, that they must be re-entrant, their
loops being closed by lengths which are ' bound ' to the aerofoil
surface, moving with it,
in the manner shown
diagrammatically in Fig.
116, all the vortex lines
being required to induce
a circulation equal to K
round the median section
of the aerofoil. It follows
that the magnitude of the
circulation round each
member of the vortex

pair is also equal to K.

Circulation applied to
a two-dimensional aerofoil
was shown in Article 135
to cause the back stagna-
tion line to be displaced towards the trailing edge. The hypothesis
introduced by Joukowski (Article 134), that for a steady state the
back stagnation line recedes exactly to the trailing edge, assumed
sharp, now receives experimental support since the starting vortex
ceases to form only when this coincidence is attained.

The aerofoil now having a circulation K midway along its span
combined with a steady forward velocity 17, it will have locally a lift
equal to pKU per unit of span. Elsewhere along the span there will
be a lift of intensity decreasing outwards because vortex lines leave
before the tips are reached, as indicated in Fig. 116. The tangential
trailing vortex sheet of Article 159 (3) is now identified with the sheet
of escaping vortex lines which continue to accumulate into further

A.D.—10*

FIG. 116. — FORMATION OF THE TRAILING
VORTEX PAIR OF A LIFTING WING.

298 AERODYNAMICS [CH.

lengths of trailing vortex as the aerofoil proceeds along its path.

No further starting vortex is formed. A picture of the vortex

system anywhere between A and H is merely an extension of Fig.

116 to include a period of uniform motion.

165. Consider a region far from the start of the flight that is

crossed by the lifting aerofoil. When the latter has progressed a

further distance, the residual
flow in the region due to
the passage of the aerofoil
approximates to a length of
vortex pair. Thus Fig. 117
shows, as an example, the
path-lines determined * ex-
perimentally 13 chords be-
hind the wing-tip of an
aerofoil of aspect ratio 6 set
at 8°. The vortex sheet

FIG. 117.— EXPERIMENTAL STREAMLINES 13 wa$ found in this case to be
CHORDS BEHIND AN AEROFOIL (THE nearly rolled UD. Again, the
AEROFOIL is SHOWN DOTTED AND ITS , „ ,/ - ~. , , _ . '
SPAN = 3 x WIDTH OF FIGURE). full line of Fig. 118 gives the

mean variation, experimen-
tally determined, of the vorticity through a wing-tip vortex well
behind an aerofoil. The dotted line illustrates the assumption made
as an approximation, viz. that vor-
ticity is uniform through the vortex
and zero in the surrounding flow.

It is easily verified that, the flight
path being horizontal, the vortices
are inclined downward by a small
angle, of the order of 1° in a practical
case, owing to their generation by
successive elements. We ignore this
angle, and, assuming a horizontal
vortex pair, enquire what lift and
drag this simplified system entails at
the aerofoil. Distance from the

starting-point of the flight and from the aerofoil permits the flow
to be regarded as two-dimensional. Let / be the distance apart of the
vortices and 2a the diameter of each. In calculating lift we neglect
the substance of the vortices, but cannot do so in calculating drag.

From Article 157 the impulse is $Kl per unit length or, since a

* Fage and Simmons, Phil. Trans. Roy. Soc.t A, v. 225, p. 303, 1925.

L_ rr-:

FIG. 118. — EXPERIMENTAL DIS-
TRIBUTION OF VORTICITY
THROUGH A TRAILING
VORTEX.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 299

length U is generated per second, the rate of change of impulse is
pX/Z7, and is directed downward. There is thus an upward re-
action L on the aerofoil, balancing the external force, given by —

JL = 9KIU ..... (207)

It is important to note that K is here the strength of the vortices.

Associated with the impulse, kinetic energy E per unit length has
been generated in the region. From Article 119 by way of the
artifice of Article 157, the impulse being constant between the vortices,
we find —

E=foK \wciy. . . . (208)

where w is the velocity of the points of application of the distributed
impulse and the integration is to extend between the vortices.
Hence —

pK>p<- / i i \

^La\¥+y ¥-y/ y

(209)

The kinetic energy Ec of the substance of the two vortex cores is
not negligible, and is derived * from the original irrotational motion
generated. Calculating it on the assumption of uniform angular
velocity —

= 2 . |p . 2n\ r . coar2 dr =

For continuity of velocity at the periphery of the cores o> = K/2na*
giving—

a result which is independent of a.

Let DI denote the contribution to aerofoil drag associated with the
continuous production of the kinetic energy of the complete residual
vortex system. D% is called the induced drag. We have, since work
is done at the rate D^U and kinetic energy appears at the rate
(E + EJU-

(211)

Induced drag is due essentially to the three-dimensional character
of the flow, vanishing for aerofoils of infinite span. It appears on

* The vortex sheet behind the aerofoil cannot contain the kinetic energy in the
cores of the developed vortex pair.

300 AERODYNAMICS [CH.

the wing as a modification of the pressure distribution appropriate to
two-dimensional flow past the wing sections at their effective inci-
dences. Hence a simpler but superficial definition is : induced drag
is that part of the pressure drag of a wing that is caused by its lift.
Another definition will become apparent in the next chapter.

To carry the foregoing expressions further and calculate the co-
efficient of induced drag for the wing, we should require to know its
dimensions besides K, I, at and U. The distribution of impulse along
the wing will in general be quite different from that which would
generate the irrotational part of the above residual motion from rest,
the vortices being spread, wholly or in part, as a sheet at the wing.
This we leave to the special investigations of the next chapter.

A first approximation to the size of the vortices may be noted,
however. It will be shown that most practical wings have induced
drags a little greater than npK2/S. Equating this minimum to (21 1)
gives I/a = 10-2. In the course of a rigorous investigation Prandtl *
obtains the value 9-2, no second term appearing in the log of (211).
A later, more physical enquiry by Kaden f suggests a mean value of
8-8. J Prandtl's result makes 2a = 0-17 X the span of the minimum
drag wing. These calculations give a fair idea of the more com-
plicated vortices found in experiment (cf. Fig. 118), and it will be
observed that their size is considerable.

1 66. Uniform Lift

Although vortex lines leave the wing along its span, forming a
trailing vortex sheet and decreasing the circulation round outboard
sections, strong vortices often exist, on the other hand, close behind
the wing-tips. These must not be confused with the residual vortex
pair, being smaller and situated farther from the centre of span. In
such cases, which are usual rather than exceptional, part of the lift
is uniformly distributed along the span, an appropriate number of
vortex lines remaining bound until the tips are nearly reached, when
they turn a corner and crowd together suddenly to form, with a
similar feature at the other wing-tip, a developed vortex pair. A
weakened vortex sheet remains between to roll up farther down-
stream. Fig. 119 (a) illustrates the system diagrainmatically.

We now treat of this uniform part of the lift as if it alone existed,
ignoring the remaining part and its associated vortex sheet (Fig.
119 (&)). The circulation K' round the wing is then constant along
the span and equal to K, the strength of each wing-tip vortex. Let

* Tragfugeltheorie. f Ing.-Arch., II, 1931.

{ Aerodynamic Theory, II, p. 329, 1935.

VORTICES AND THEIR RELATION TO DRAG AND LIFT

VII]

2s be the span, c the chord, and A the
aspect ratio = 2s/c, and as before let / be
the distance apart of the vortices, each of
diameter 2a. According to experiment I
is slightly less than 2s, but for simplicity
we ignore the difference, and also, in
calculating lift, the vortex diameters.
Then we have for the lift on the wing —

L =K'pU .2s . . . (212)

and for the rate of change of impulse
required to generate the vortex pair con-
tinuously :

dl/dt = KpUl,

expressing equality if K' = K and 2s = /.
of vortices remains at 2s apart,
longer necessary —

r - L 2K

CL =

301

(a)

FIG. 119.

We assume that the pair
Omitting the accent in (212) as no

and then from (211) —

2sc

0-8

FIG. 120. — EXPERIMENTAL RELATIONSHIP
BETWEEN THE LlFT OF A R.A.F. 15
AEROFOIL AND THE IMPULSE OF THE
VORTEX PAIR EXISTING 2 CHORDS BE-
HIND IT.

(213)

(214)

Fig. 120 gives the results
of experiment * with a thin
aerofoil of the section known
as R.A.F. 15 and A = 6 at
a Reynolds number of 6-3
X 10*. The measurements
were made at a distance of
2c behind the aerofoil, but
little difference resulted
from considerably reducing
or increasing this distance.
At flying incidences ap-
proximately 75 per cent, of
the lift was uniform, while
this percentage increased at
larger incidence, past the
critical. A thick aerofoil of
deep camber and the same

* Piercy, Jouv. Roy. Aero, Soc., October 1923.

302

AERODYNAMICS

[CH.

aspect ratio showed 50-60 per cent, of the lift to be of this kind at
8° incidence. A slightly weighted mean of c/a was 13, or we may
take 2s /a = I/a =78. If CL, CD> refer to the uniform part of the lift,
we find from (214) —

CDi = 0-061 CL« . . . . (215)

For future reference it is convenient to express this result in the
form —

(1+0-15).

• (216)

Induced Dra9

ntre of Span Wing tip — *|

FIG. 121. — DISTRIBUTION OF INDUCED
DRAG FOR UNIFORM LIFT.

The induced drag for uniform lift is distributed between the vortices
in the same way as the induced velocity of the vortex pair (Fig. 121),

and since 2s =/ and the vortex
pair forms immediately, we infer
the same distribution along the
aerofoil. It is a minimum at the
centre of span, where the pressure
distribution will most nearly ap-
proach that for two-dimensional
flow, and increases rapidly towards
the tips. Lack of knowledge of

| ' 1 the curl and spread of the vortices

|«— Centre of Span w&ng tip — *| at the tips prevents completion of

the figure, but small areas of
highly reduced pressure are com-
monly found here on the back

part of the upper surface of an aerofoil. A great advantage of
aspect ratio in reducing induced drag becomes evident when it is
reflected that, for greater span, lift increases without increase of K.
Aerodynamic calculations involving a knowledge of the degree to
which the vortex sheet has rolled up, e.g. on tail-setting angle, are
complicated, and the assumption is sometimes adopted that the whole
of the lift is uniform. Alternatively we may assume that the
residual vortex pair is developed quickly, as tends to occur at large
incidences.

167. Variation of Circulation in Free Flight

The argument of Article 164 is readily elaborated to include varia-
tion in the velocity of the aerofoil. If, after a period of steady
motion, the velocity is increased, the original circulation becomes
insufficient to keep the back stagnation line on the trailing edge and
a new starting vortex is thrown off during the time of acceleration.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 303

This joins together additional vortex lines packed into the trailing
vortices, which are strengthened thereby, and increases the circula-
tion round the aerofoil. A decrease of velocity produces the opposite
result, a retardation vortex leaving the aerofoil to close vortex lines
no longer required in the weakened trailing vortices appropriate to
the reduced circulation. Such a sequence of events requires suitable
variation of the external force which constrains the aerofoil, whose
incidence is assumed constant, to move in a straight path in spite of
variation of lift.

The same principles apply, of course, to the wings of an aeroplane
in horizontal flight, but the argument needs modification to take
account of the fact that the downward component of the constrain-
ing force applied to the wings is equal to the sum of the total flying
weight and any downward air loads on other parts of the craft, and is
approximately constant ( = W, say).

When the velocity of an aeroplane is increased from U to [/',
we must have (neglecting variation of /) in order that flight may
remain horizontal —

pK'U'l = ?KUl = W

°r~ K'jK = U/U' . . . (i)

i.e. variation of speed requires inversely proportional variation of
circulation. This is secured (considering increase of speed) by such
a decrease of angle of incidence as will, during the change, tend to
move the back stagnation line rearwards to a greater extent than the
acceleration tends to move it forwards, so that the vortex thrown off
from the trailing edge is opposite in hand to a starting vortex.

In terms of the lift coefficient, we have, for steady horizontal
flight—

CL'ipt/"S = CLiptf«S = W,
where S is the projected area of the wings, or, since this is constant — •

Q//CL = (U/Uy. . . (ii)

Taking for simplicity the case of uniform distribution of lift along the
span and constant chord, we have, from (213) —

K' - CL/[//

'K~~Cjfi

agreeing with (ii) on substitution from (i).

When flight has lasted for an appreciable time, the vorticity of the
original starting vortex will have diffused, through the action of
viscosity, and as time proceeds, length after length of the trailing
vortex pair far behind the aeroplane will similarly diffuse.

304

AERODYNAMICS

[CH.

1 68. Example from Experiment

The following analysis of some experiments * with an aerofoil in a
wind tunnel illustrates (1) approximate allowance for wall constraint
(technical conversion formulae are developed in the next chapter),
(2) application of the simplified vortex configuration, (3) the Rankine
vortex assumption.

The deeply cambered rectangular aerofoil, 2 ft. span and 0*33 ft.
chord, was suspended symmetrically at 8° incidence in an enclosed-
type tunnel 4 ft. square in section, the undisturbed air speed being
31'3 ft. per sec. Downwash angle was explored by means of the
meter shown in Fig. 122, consisting of two fine tubes inclined at 46°

30°

20°

u.
O

10° a

|*CENTRE OF SPAN

~3c -2c -c

DISTANCE BEYOND WIND -Tip

FIG. 122. — MAXIMUM DOWNWASH ANGLBS OBSERVED 2 CHORDS BEHIND AN AEROFOIL
BY MEANS OF THE METER SHOWN INSET.

to the stream, with their open mouths touching. This was mounted
on a cranked arm turned by a micrometer wheel outside the tunnel
about a centre-line passing through the point of contact. The
instrument could be traversed parallel to the trailing edge of the
aerofoil. The aerofoil could be traversed parallel to its lift. Thus,

* Piercy, he. cit.t p. 227.

VORTICES AND THEIR RELATION TO DRAG AND LIFT

305

VII]

after calibration of the tunnel with the model removed, downwash
angle could be measured by orienting the meter to give equal
pressures in the two tubes at any point at a set distance behind the
aerofoil. This distance was 0*9 ft. behind the centre of pressure of
the lift, through which the bound vortex lines are assumed, in the
subsequent analysis, to be concentrated.

Observed downwash angles are given in Fig. 122 for a line nearly
level with the trailing edge, but adjusted to give maximum slope to
JM. A vortex of 0«05 ft. diameter with centre at J and of approxim-
ately uniform angular velocity is clearly indicated. The distance /
between this and the corresponding vortex behind the other wing-
tip was found to be 1-89 ft.

Constraint by the floor and roof on the downwash from the bound
vortex AB (Fig. 123) will be calculated from Article 155. Actually,

FIG. 123.

the constraint will be less owing to the short length of the aerofoil,
but this correction is only 10 per cent, of that for the side walls and
only 1 per cent, of the mean downwash, so that precision is unneces-
sary. Thus, from the figure, if wl is the downwash velocity from the
aerofoil at P distant y from BC :

K , -v/j.

wl = - — (cos a + cos £

x being distance behind AB. The last factor = 0-92.

Allowance for constraint on the wing-tip vortices will be made by
substituting a tunnel wall of circular section, a radius of 2-1 ft. being
chosen for reasonable coincidence with the square wall

The two imaged vortices AD', B'C' are distant (2-l)*/i(l'89) =
4-67 ft. from the centre of the tunnel, and lie in the plane containing
the real vortices. Thus, on account of the trailing vortex system,
there is at P :

(a) An upwash velocity — wt due to BC given by —
K .

306

AERODYNAMICS

[CH.

(b) A downwash velocity wa due to AD given by-

K (cos S + 1)

w, =

(d) An upwash velocity — w6 due to A'D' given by —

The total downwash velocity is given by —

w = wl —

DISTANCE OUTBOARD FROM \MWO-TIP

FIG. 124.

VII] VORTICES AND THEIR RELATION TO DRAG AND LIFT 307

Evaluating w for a series of values of y gives the curve of Fig. 124,
for which the value 4-0 has been chosen for K. The points marked
about the theoretical curve are experimental, and are derived from
the readings of Fig. 122 by assuming the horizontal component of
velocity to be unchanged beyond the wing-tip. The ' fit ' of the
curve to the observations is better near to and far from the vortex
than it is at intermediate positions, but upon the whole it is fair, and
K = 4-0 is justified.

To obtain an independent check on this value, measurements were
made of the pressure within the vortex at a number of radii. These
are shown as points in Fig. 125, together with the theoretical curve

FIG. 125. — VARIATION OF STATIC PRESSURE THROUGH A TRAILING VORTEX.
The curve is theoretical, the circles are observations.

obtained from (179) and (180) with K = 4 and a = 0-025 ft. as
measured. It will be seen that the check was successful.

Let us now calculate the down wash velocity w' at the point F mid-
way between the vortices and 0-9 ft. behind the centre of pressure of
the aerofoil. Denoting AF = BF by mt the expression is —

(4-67)*}

where the first term gives the contribution from the circulation round
the aerofoil, assumed constant and = K ; the second that from the
trailing vortices ; and the third that from their images. In round
numbers this reduces to —

308 AERODYNAMICS [CH. VII

K. K

w' = — (1-48 -f 3-58 — 0-51) = — X 4-55

4?r 4rc

= 1»45 ft. per sec.

If the horizontal component at F were 31'3 ft. per sec., the angle of
downwash there would be 2-7°. A considerably greater value than
this is expected owing to the reduction of the horizontal component
in the wake, but not nearly so great a value as measured, viz. 6-1°.
Hence, clearly, all factors have not been taken into account. This
was immediately evident on weighing the lift of the aerofoil, which
came to 1*61 KpUl. The conclusion is that so close behind this
aerofoil 40 per cent, of the vortex lines escaping into the wake had
not yet rolled up into the vortex pair.

Chapter VIII
WING THEORY

169. The present chapter studies in more detail wings of the
strictly limited span practicable for sustaining heavy flying loads.
The boundary layer is supposed everywhere to be very thin, and skin
friction and the viscous wake are neglected. It follows that form
drag is assumed to be zero and the incidence sufficiently removed
from the critical angle.

Introductory articles on the theory of the lifting monoplane have
described the residual flow caused in a region of the atmosphere far
behind the wing as consisting of a vortex pair, and some detailed
investigation has been given to a simplified vortex configuration in
which vortices are conceived to spring from the wing-tips, necessitat-
ing uniform lift along the span. We now proceed to examine the
flow close to the wing with a view to investigating wing forms.

Fluid velocities are compounded of the translational velocity and
a component due to the complete vortex loop, part of which is bound
to the aerofoil and produces circulation round its sections. Exact
calculation of the second component would be complicated, and
would involve more precise knowledge of the three-dimensional
distribution of the vorticity than is available. An approximation,
appropriate to calculations in the vicinity of the wing, is at once
suggested, however, by theory and experiment. Chapter VII clearly
indicates that at a point close to a system of vorticity the velocity
will be affected much less by distant parts of the system than by near
parts. On the experimental side, the vortex sheet spreading behind
the aerofoil has been found in some cases to roll up only slowly,
apart from discrete vortices immediately formed. Consequently, a
reasonable approximation for present purposes is to regard the free
vortex lines as trailing behind the aerofoil perpendicular to its span.
This parallel formation cannot extend indefinitely, as we have seen,
but the form of (182) permits us nevertheless to regard without impor-
tant error the straight vortex lines as being of semi-infinite length.

The monoplane wing is completely represented in the present
investigations by its span, aspect ratio and the span-wise distribu-

309

310 AERODYNAMICS [CH.

tion of circulation round its sections, i.e. the grading of lift intensity.
Both these quantities are at the choice of the designer within certain
limits. Good approximations to a desired lift distribution can be
obtained with different shapes of wing to meet other requirements.
The scope of the enquiries is expressed in the following set of
general equations, whose construction will now give no difficulty,
but whose solution in a given case may be attended with analytical
complication. The peculiar nature of the results will fortunately
enable us to avoid much of the latter when small errors can be
tolerated.

170. General Equations of Monoplane Theory

Take the origin at the centre of span (Fig. 126), and denote by K
the circulation round the wing at a distance y towards the starboard

FIG. 126.

wing-tip.

J TS

The circulation at y + 8y is K + — 8y.

It is assumed

that the circulation diminishes outwards from a maximum J£0 at the
centre of span. Hence, over the element of span Sy, vortex lines to

the strength — — Sy leave the aerofoil to form part of the vortex
ay

sheet, and the direction of rotation is such as to cause a downwash
nearer the centre of span. Denoting the velocity of this downwash
at yl along the aerofoil by 8wlf we have from (184) and by Article 169 —

Trailing vortex lines arise in this way all along the span, and
produce a component of the downwash velocity, which is in general
variable along the span. Its value at any point is due to all the
vortex lines not passing through that point. To calculate the value
wl at yl we have, from (i) —

VIII] WING THEORY 311

2s being the span. The integrand approaches oo as y approaches ylt
so that integration must be stopped short of yl and the limit investi-
gated. This kind of difficulty occurs frequently in our subject ; it
can usually be met by considering elements close to yl on either side
in pairs. (See also Article 140D.)

Consider the circumstances of the element at y. Its reaction
amounts to pKq . Sy, q being the resultant of w and the translational
velocity U. This reaction is perpendicular to the local relative
motion, and is therefore inclined backwards from the perpendicular
to U and to the span, i.e. from the direction of lift. The element SL
of lift is—

8L = ?Kq . Sy . U/q = ?KU . 8y. . (ii)

There is also an element SZ)^ of induced drag, i.e. a component
parallel to C7, given by —

w
SA = ?Kw . Sy = ~ 81 . . . (iii)

For the whole aerofoil —

L = p£7 Kdy (218)

J — s

Kwdy (219)

Again referring to the element at y, another effect of w is to reduce
its incidence from a0, the incidence it would have consistent with its
lift if it formed part of a wing of infinite span so that w vanished, by
the angle w/U, assuming this to be small. Hence, to realise the lift
(ii), its incidence, measured from the angle for no lift, must, com-
pared with two-dimensional conditions, be increased to a according
to—

a = a0 + w/U .... (220)

171. The above alternative theory of induced drag must give the
same results as that described in the last chapter. This is readily
verified in the case of uniform lift, when the whole of the trailing
vorticity is gathered together at the wing-tips as a vortex pair,
provided we exclude a region at each tip where w cannot be deter-
mined and K is indefinite.

Let I be the distance apart of these vortices and 2a the diameter of
each. At any position y along the span of the aerofoil —

„-*( l , 1

312 AERODYNAMICS [CH.

provided y <\l — at say, where a is the extent inwards from each
vortex centre of each of the excluded regions. Hence, from (219) —

1 1

= p - log ( -- 1

*27u 6 \a

agreeing with the induced drag evaluated over a corresponding length
of the aerofoil by the method of Article 165.

172. The ' Second Problem ' of Aerofoil Theory

We now approach the question of the distribution of a given total
lift over a given span for minimum induced drag. This problem is
evidently of the greatest engineering interest, provided that the
distribution found can be realised in a wing which is structurally
economical in weight. Investigation is possible in several ways. To
introduce the method developed below, the problem may be restated
as follows.

A wing traversing a region of the atmosphere at constant speed U
and small positive incidence — say horizontally — exerts on the air a
rate of change of impulse in a downward direction. Let / be the
impulse imparted per unit length of the flight path. We have —

L=IU ..... (221)

In the process, kinetic energy is communicated to the air at the rate
E per unit length of the path. The whole flow is regarded as irrota-
tional, the substance of the vortices formed being neglected ; work is
done by the impulsive action at the rate EU and, as in Article 165,
we find Dv = E. Total lift and span being fixed, what con-
ditions will make E a minimum ? What form of wing, if any,
will realise these conditions ?

173. Distribution of Given Impulse for Minimum Kinetic Energy

Consider first the sum of two impulses Ilt /2, regarded as acting
on fluid masses mlt ma, and changing their velocities from Wlt Wt
to wlt wt. By Article 119 —

I, = m, (w, - WJ, /, = mn (w. - W.).
It is assumed that the sum 7t -f Js = const. = C, say, or —
C = mlwl + m^w2 — (m^Wi + m^Wt).

The work done by the total impulse is given by the change E of
kinetic energy, and we have —

VIII] WING THEORY 313

Therefore —

-- (C + mlWl + m*W2)*

w2w22) — (mlwl + waze>8)a
= mlm2 (z£>!2 + z0aa — 2wjW2)

The last expression, being necessarily positive or zero, has a
minimum value when wt = w2. Now all the terms on the left-hand
side of the equation are known and prescribed except 2E. Hence,
£ is a minimum when wl = w2, i.e. when, whatever the initial
velocities, the final velocities are equal.

The corresponding result for a number of impulses Ilt I2, 78 . . .
is proved in exactly the same way. Let these act on masses mlf m2,
ws . . ., changing their velocities from Wlt W2t W% . . ., to wlt w2,
w9 . . ., and let —

1 1 + 1 2 + h + • • • = wt (w1 - Wl) + m2 (w2 - Wi)

+ m> (w, -W9)+ ...
= C, a constant.
We have —
2E = ml (wf - Wf) + m2 (w2* — PF22) + ms (w^ - WJ) + . . .

and, therefore, writing EQ for the original kinetic energy and M0 for
the original momentum —

(Lm) (2E + 2E0) - (C + M0)'

— (mlwl + m2w2
= ml (m2 + mn + . . .) w£ + m2(ml + m* + . . .) wf

+ w3 (w! + m2 + m< + . . .) wj + . . .

— 2(m1m2w1w2 + m^m^w{wz + . . .)
= mtm2 (wt — w2)* + w^s (te^! — w8)a + • • •

The right-hand side being necessarily positive or zero, the left-
hand side is a minimum when wl = w2 — z#8 = . . . Therefore
£ is a minimum under this condition.

We conclude that a distributed impulse of given total magnitude
does minimum work when the final velocity of its point of applica-
tion is everywhere the same.

174. The foregoing condition for minimum work done is realised
when the impulse is applied through a rigid plate accelerated normally
to its plane.

Considering a region of initially stationary air which has been
traversed by a wing, the kinetic energy in unit length parallel to the
flight path will be a minimum for a given lift and span, if the flow

- - sin

314 AERODYNAMICS [CH.

there is such that it might have been generated from rest by the
acceleration in a normal direction of unit length of a long plate of
width equal to the wing span. Two-dimensional conditions are
appropriate, the wing having passed ahead. / and E are obtained
immediately from Article 120. If w is the final velocity of the plate
and 2s its width —

/ = TTpte'S1 ... (i)

E = ^Trpw's1 (ii)

The distribution of / across the plate is shown in Article 120 to be
elliptic.

On the plate <f> is given by —

^ = — . ws sin 7) . . (iii)

7) ranging from 0 to 2?r round the surface. Considering two adjacent
points A and B, midway between the edges and on opposite sides of
the plate, the interval of </> between them is —

37T\

== 2ws .... (iv)

Now actually, the plate so far imagined is fluid, consisting of a
vortex sheet, and this interval of <£ is the circulation round any
circuit passing through the centre line and embracing one-half of the
sheet. Denote this circulation by K0. From (iv) —

KQ = 2ws
or —

w = K0/2s .... (222)
Substituting in (i) and (ii) —

I = P™K, .... (223)
*-Pg*.' .... (224)

175. Elliptic Wing-loading

In applying this important result to the actual case, when the
motion is generated by a wing of lift L and span 2s, we note the
following :

K0, being the circulation round each half of the vortex sheet,
must also be the circulation round the median section of the aerofoil.
Hence the central intensity of the lift is pKQU. At outer sections

VIII] WING THEORY 316

this is required to fall away elliptically, i.e. the lift intensity at y is
given by —

. (225)

We note —

= p-K.U (226)

agreeing with (223) from (221).

The velocity w has, at the aerofoil, from Article 148, one-half its
value far downstream given by (222), i.e. at the aerofoil —

w = K0/4s (227)

Knowing this velocity to be constant along the span, we can readily
recalculate its value from (217). From (225) —

dK K, y

;£--T^=- • • - <»')

Hence, applying (217) to yl = 0 —

K0 f5 dy Ko

w = — - J — -. = -

Again, since w is constant along the span —

~ ^T TC_

(229)

from (226) and (227), in agreement with (224), remembering that
E = Dt. Substituting for K0 from (226) and writing X for the
span-loading L/2,s gives —

Di = 2XV7ip?7a . . . (229a)

The best wing shape which, according to theory, should give the
required uniform induced velocity at itself, is of elliptic plan-form,
having geometrically similar sections, so that camber is constant,
set at constant incidence along the span. This shape is quite practic-
able in modern wing construction. Other plan-forms can be
arranged to give approximately elliptic loading by suitably gradating
camber and incidence along the span, but these may hold only for a
single value of the central incidence.

316 AERODYNAMICS [CH.

Criticism of the feasibility of realising the minimum of induced
drag arises actually on the theoretical side. Considering again the
thin flat vortex sheet, (228) shows that dK/dy tends to oo at the wing-
tips. Essentially the same result follows alternatively from Article
124 ; on analogy with a plate in broadside-on motion, (153) gives the
span- wise components of velocity over the faces of the sheet, which
tend to oo at the edges. The form of vortex sheet calculated could
not persist at its edges and modifications are to be expected near the
wing-tips. On the other hand, we shall see shortly that (229) is not
a critical minimum. Hence, while the minimum induced drag
may be regarded as an ideal impossible to realise exactly, small
departures from the conditions required cause increases which for
some practical purposes are not important.

176. Minimum Drag Aerofoil Formulae

Let A be the aspect ratio of the wing giving elliptic span-loading.
For a rectangular plan-form of constant chord c, A — 2s/c. If the
plan-form be shaped, having area S and mean chord c' —

2s 4sf 4s1

x-?-iT?=T ..... (230)

From (226)—

L~iPC7'S~pC7* 4s» 2 U 2s* *

This may be compared with the result (213),CL = 2K/Uc,ior uniform
lift and constant chord.
From (229)—

A _2D, A _n X0» ^ _ 1
D'~~''"~~* ' ~ L '

by (231). Similarly for the k system of coefficients

Formulae (227) and (231) yield—

w I

„ _JL_ r * (9w\

TT ~~~~ A L * * * * ^*iOOy

Substituting in (220)—

a = a0 + ~ CL . . . (234)

These formulae are often employed to calculate changes conse-
quent upon modification of aspect ratio. For this purpose we have —

VIII]

WING THEORY
1/1 1

317

„

' 1(l l\r

-« =«(A-A')C»

By differentiation of (234) —

da. d<x.Q I I

~ 1 'X~» I jt ~ ft

. (235)

if the theoretical slope 2?r for thin aerofoils of Joukowski shape be
accepted. Then —

-"§-+:

. (236)

An empirical correction is
to put dCLjd<x.Q = /.27r, r

where / = 0-87 approx. from
experiment. Then dCJdcx. =

f\ A I tc\ I A I f\

177. Examples

The foregoing simple
results are of outstanding
practical importance. They
are often known as ' reduc-
tion formulae/ The follow-
ing examples, illustrating
their many applications, rest
upon certain assumptions ;
e.g. the difference between
elliptical and rectangular
plan-forms is for the time
being regarded as negligible.
These are discussed after
further development of the
theory.

(1) The extended curves
of Fig. 127 give the experi-
mental * lift and drag co-
efficients at a full-scale
Reynolds number for a rectangular wing of aspect ratio 6 and of

* Experimental data in this article are based on Relf, Jones, and Bell. Loc. cit.t
p. 91.

O°

10°

FIG, 127.

318 AERODYNAMICS [Cfi.

the section shown in Fig. 42, known as R.A.F. 38 (thickness ratio
== 0-127). Consider the effects of increasing A from 6 to 12. Im-
mediately from (235) — AC^. = CL*/l2n, — Aoc = CJl2n. Adding
these increments to the experimental values of C^ and a correspond-
ing to any value of CL gives one point on each of the derived curves
for A = 12 shown. It will be noted that contributions to CD of
form drag and skin friction, necessarily included in the experimental
values, are left unchanged. This will be shown to be justifiable
within certain limits ; the large decrease in drag at appreciable
lift coefficients due to doubling the aspect ratio would be realised

in practice.

(2) Fig. 128 shows CD$
plotted against CLfor elliptic
wings of aspect ratios 6 and
9, and also the minimum
total CD that can be expected
for a rectangular wing shape
of aspect ratio 6 and thick-
ness ratio = 0-12.

At moderately large lift
coefficients and incidences,
but appreciably below the
maximum lift stage, induced
drag is much greater than
the sum of form drag and
skin friction, and increase of
aspect ratio produces marked

improvement. The advantage of high aspect ratio diminishes,
however, with small lift coefficients. For instance, at CL =1-0 in
the present example, C^. forms nearly 80 per cent, of CD, and
increase of A from 6 to 9 would save 25 per cent, of the whole drag.
But at CL = 0-2 the induced drag is only 20 per cent., and the
possible saving would amount to only 7 per cent.

Consider a monoplane using this wing, of 5 tons total weight and
having a minimum flying speed of 60 m.p.h. At a cruising speed of
110 m.p.h., about 180 b.h.p. would be absorbed by the wing alone,
assuming an airscrew efficiency of 80 per cent. Of this, 18 per cent,
could be saved by increasing A from 6 to 9. But a speed of 180
m.p.h. would require 450 b.h.p. for the wing, and a decrease of only
4 per cent, could be achieved.

It must be borne in mind that structural questions qualify the
advantage of high aspect ratio. Full cantilever construction would

OO4 0-08

Cj) AND

FIG. 128.

0-16

VIU] WING THEORY 319

be too heavy, with the materials at present available, for so thin a
wing as that considered with -4=9, and the smaller induced drag
would be offset by the added drag of external bracing. A thick
wing may be substituted to overcome this difficulty though form
drag increases. But a point appears in favour of tapered plan-form
which reduces the bending moment at the root of the wings.

(3) A monoplane weighing 9 tons has wings of aspect ratio 7 and
780 sq. ft. area. At 5000 ft. altitude its maximum rate of climb
occurs at an indicated airspeed of 140 m.p.h., when CL = 0-6, and
is 1080 ft. per min. Determine the aspect ratio for new wings,
neglecting their increase in weight, to increase the rate of climb by
5 per cent.

The additional thrust h.p. that must be made available for
climbing is —

0-05 X 20160 X 1080
33000 = 33'°-

At 5000 ft. pl7* = 100-3 and U = 205-3/^0-862 =» 221-2 ft. per sec.
If A is the new aspect ratio which will lead to a saving of 33 thrust
h.p.-

-U — 4 MO'6)1 X i X 100-3 X 221-2 X 780 = 33 X 550

or —

A = 8-03.

The above data relate to a modern craft with a usual minimum
flying speed, with split flaps fitted to the wings, and whose cruising
speed would be about 200 m.p.h. Let us change them to apply to
a slow craft. It is only necessary to change the speed at which
maximum rate of climb occurs from 140 m.p.h. to 100 m.p.h. The
required increase of aspect ratio is then found to be from 7 to 10-8,
owing simply to the reduction of speed, lift coefficient being supposed
kept constant by reduction of minimum speed.

The foregoing examples illustrate that high aspect ratio substanti-
ally improves the speed or economy of aeroplanes of restricted speed
range, and that it benefits aeroplanes of large speed range chiefly in
regard to rate of climb and ability to maintain altitude when only
part of their power equipment is functioning.

(4) A disadvantage of high aspect ratio arises in some cases as
follows :

Wings must be made sufficiently strong to withstand the shock of
flying into an upward gust, which has the effect of increasing inci-
dence suddenly and thus momentarily increasing lift. The following

320 AERODYNAMICS [CH.

calculations are based on 25 ft, per sec. for the upward velocity of the
gust, which is not excessive.

Assume the wings to have an effective maximum CL of 1-28 and to
be of aspect ratio 6 or 10, alternatively ; dCJdv, is then either 4-712
or 5-236, theoretically.

Consider first a craft of 60 m.p.h. minimum flying speed encounter-
ing the gust at 120 m.p.h., so that initially CL= 1-28(60/120)*= 0-32.
The sharp increase of incidence comes to 25/176=0-142 radian
and CL increases by 0'670 for A = 6 or by 0-744 for A = 10,
becoming 0-990 or 1-064. Thus until incidence and speed have
time to change, the lift of the wings is increased in the ratio 3-10
or 3-33, approximately. The wing of the higher aspect ratio is
the more severely stressed by the gust. Let us diminish the mini-
mum flying speed to 50 m.p.h. and advance the speed range to 3.
Then initially CL =0-142, Aoc =0-114 and CL becomes 0-680 for
.4=6 and 0-740 for A = 10. The transient ' load factors ' are now
4-78 and 5-20, respectively. A factor of safety of 2 is usually called
for, and this excessive loading might provide an overriding condition
for the structural design of the wings, which would therefore be
heavier with the larger aspect ratio on a score additional to that
mentioned in (2).

178. The Arbitrary Monoplane Wing

The problem of the wing of non-elliptic form, defined by a given
variation along the span of shape of section, incidence a and chord ct
is somewhat complicated. A brief description is given below of a
convenient method * of solution developed by Glauert, whose book f
should be consulted for further details. As before, the area of the
wing is written S and its span 2s.

Change the variable so far employed to denote position along the
span from y to 6, according to —

y = — s cos 6.

Then 6=0 when y = — s, i.e. at the port wing-tip, 0 = n/2 at the
centre of span, 0 = TT when y = s, and dy = s sin 0^0.

The circulation K, varying along the span, is expressed in a series
of Fourier type —

K = 4sC7SQ sin nQ (237)

* An alternative method is due to Lotz ; see Shenstone, Jour. R. Ae. S., May
11)34.
t The Elements of Aerofoil and Airscrew Theory.

VIII] WING THEORY 321

in which for symmetry about the median plane only odd integral
values of n appear. Since from (218) —

substituting from (237) and remembering (230) gives —
CL = 2 A p (ECrt sin n8) sin MQ,

where A is the aspect ratio. This is easily evaluated, giving —

CL==nAC1 .... (238)

The result means that the first coefficient of the Fourier series
determines the total lift coefficient. It does not mean that CL oc A.
C8, C5, . . . modify the distribution of lift along the span, but in a
manner which leaves CL constant. This distribution depends upon
the shapes of the sections, their sizes and attitudes all along the span.

For an exact solution the summation should be from 1 to oo , or at
least include a considerable number of terms, each relating to a
particular outboard position, i.e. to a particular value of 0. But the
Fourier series often gives a good approximation when only few terms
are used ; the practical feasibility of the method depends upon this,
and usually four or five terms are sufficient when, from symmetry,
only the semi-span need be considered.

We have to find an equation for Cn which will be satisfied every-
where along the span, although in practice it will be satisfied at a few
chosen positions only. In framing the equation we have to relate K
to the induced velocity w, now also variable, according to the funda-
mental laws developed in Article 170 and Chapter VI.

At any position 0 —

29KU =CL6pcU>,
or —

2K = CL9cU.

Also, if cLQe is the incidence under two-dimensional conditions
measured from the angle of no lift —

It is convenient to ignore for the moment variation of the slope of the
two-dimensional lift curve from one section to another and to write
for this 2?r, its theoretical value for thin Joukowski shapes. Then —

322 AERODYNAMICS [CH.

and hence, dropping the suffix —

K = Ttt/coco
or from (220)—

K - rct/c (a - ~). . . . (239)

Now, on substituting the new variable and taking account of
(237), (217) can be reduced (cf. Article 140D) to—

. . . . <*.)

Substituting this expression for w and also for K in (239) —

A vr • Q / ^nC* sin n®\

4sSCrt sin n6 = nc [ a -- 7— - - ),
\ sin 6 /

or —

EC, sin nQ (~ + sin 6 ) = ~ oc sin 6 . . (241)

\ 4s /4s

This is the general equation required. More accurate values of
dCJda. than 2:r will be known for the sections under two-dimensional
conditions at the chosen 0 points, and these should be substituted in
practice for 27c. But we shall retain the approximation. With it,
results are obtained in terms of the parameter c/4s, which is in
general variable, but which is constant and equal to 1/2.4 for a given
rectangular wing.

We are now in a position to calculate the total induced drag.
After substituting for y, K and w, (219) leads to —

= 2A (SnC* sin n6 sc sin n d®'

and this reduces to —

Cw =7i4SnC/. . . . (242)

179. The ElHptically Loaded Wing Compared with Others
By squaring (238) and substituting in (242) we find—

• • • <243>

Now the sum is obviously positive, and CL is specified by d.
Therefore, the induced drag is a minimum for a given lift when all
coefficients subsequent to the first vanish. The sum will then

VIII]

WING THEORY

323

reduce to unity, and thus the minimum possible drag coefficient for
any wing is given by (232), as already proved, when lift varies
elliptically along the span.

The formula (243) is conveniently written—

(244)

For elliptic loading 8 = 0, but for other distributions it has a positive
value — a small fraction in practical cases, as will be illustrated. 8
then gives the proportionate increase of induced drag above the
minimum theoretically possible.

Aeroplane wings are commonly, for constructional reasons, either
rectangular or straight-tapered, except for rounded tips. With
square tips, these plan-forms have been investigated by Glauert,*
using the method just described, and the rectangular shape by Betz f
and others, employing different methods. Some load distributions

FIG. 129. — MATHEMATICAL DISTRIBUTIONS OF LIFT FOR VARIOUS PLAN-FORMS.

are illustrated in Fig. 129 for equal total lift ; uniform loading is
included, although this is not a practical, nor a desirable, case. The
results relate to constant shape of section and geometrical incidence
along the span. The half-taper wing has a tip chord equal to
one-half its central chord, and its loading approximates to elliptic
loading. The distribution for a much sharper taper differs as
much as does that for the rectangular shape from the ideal,
but in the opposite way. Both these loadings differ widely
from the elliptic considered from a structural point of view, but
the question remains as to how different they may be as regards
induced drag.

*Loc. cit.t p. 320. t Dissertation, G6ttingcn. 1919.

324

OO8

AERODYNAMICS

OO6

O02

FIG. 130.

[CH.

Fig. 130 indicates the
theoretical variation of 8
with A for rectangular
wings according to Glauert
(full line) and Betz (broken
line), showing good agree-
ment. The slope of the
two-dimensional lift curve
is assumed to be 2n ; if it
is less, aspect ratio should
be proportionately increased.
Induced drag decreases with
increase of At but not so
quickly as for the elliptic
wing. This is illustrated in
Fig. 131, where also are

shown the result estimated from experiment in Article 166 for

uniform lift with ^4 = 6, and a theoretical result for a pointed wing.

The theoretical error in estimating induced drag for rectangular

wings by (232), i.e. by assuming elliptic loading, decreases from 8 per

cent, at A = 10, which

is a large aspect ratio 0-08

for aeroplane wings,

to 5 per cent, at A =

6-7, and to 3 per cent.

at the aspect ratio 4

often employed for tail- £

planes.

Fig. 132 gives the

variation of 8 with taper

for A = 6-7. The best

taper, from the present

point of view, has a tip

chord somewhat less than

one-half the central

chord at this aspect

ratio. Induced drag is

then only 1 per cent.

greater than for elliptic

loading. Structural

questions may suggest

a sharper taper, but

006

2
L

004

002

SHARP TAPER
RECTANGULAR

FIG. 131. — INDUCED DRAG AND ASPECT
RATIO.

(* This point is estimated from experiment.)

WING THEORY

325

008
OO6
OO4
OO2

VIII]

8 then increases until, with
a pointed wing, induced
drag becomes as great as
has been estimated for uni-
form lift [cf. (216)].

The conclusion is that the
induced drag of normal types
of wing, of moderate aspect
ratio and taper, can be
assessed to a good first
approximation by the form-
ulae of Article 176, and
especially changes due to
modifications of aspect ratio
provided the type of loading

remains the same. More accurate reduction formulae than (235)
are, however, easily deduced by precisely the same method.

Increased induced drag implies that, to secure a specified lift
coefficient, incidence must be increased more than is provided for
by (234). For a full discussion of this question reference should be
made to the original papers. If, on analogy with (244), we write —

O OZ 04 CK> O8 1

TIP CHORD+CENTRAI CHORD

FIG. 132.

- a0 + - CL (1 + T),

(245)

T increases approximately linearly for rectangular wings from 0-1 at
A = 3 to 0-23 at A =10, Associated with this change is a decrease
in the slope of the lift curve, as compared with two-dimensional
conditions, greater than is calculated in (236). For rectangular
aerofoils, the lift curve of whose section has a slope 2n in two-dimen-
sional flow, the effect is approximately allowed for by substituting
3-02 for 7t in (236).

1 80. Comparison with Experiment

The solution for the rectangular wing provides convenient means
for comparing the results of aerofoil theory with experiment, since
aerofoils of this shape can easily be made with accuracy, and many
checks of different kinds have been obtained.

The slope of the lift curves for R.A.F. 38 and Clark YH aerofoils
with A = 6 are 0-0752 and 0-0742 per degree. A number of good
aerofoils give a slope rather less than 0-076 at this aspect ratio and
at fairly large Reynolds numbers. The theoretical slope for a two-
dimensional slope 27i is 6-04 x 6/8 = 0-453 per radian = 0-079 per

326 AERODYNAMICS [CH.

degree. It appears that at full scale a less factor than 3-02 should be
used in place of n in (236), but the agreement is nevertheless quite
good.

Comparisons between load grading curves are less satisfactory
except at very small incidences. Fig. 133 shows as a full line the
experimental distribution along the span of a rectangular aerofoil

obtained by integrating the
pressures over the surface, and
as a dotted line the theoretical
result. A marked difference will
be seen to occur towards the
wing-tips. The angle of inci-
dence was 6° ; the difference is
typical for incidences greater
than about 1°, and occurs also
100 _ on tapered aerofoils. The

FIG. 133. — DISTRIBUTION OF LIFT , . ,

ALONG RECTANGULAR WING. pressure peak near the tip sig-

Full line : experimental ; broken line : nalises a knuckle of hoarded
theoretical^ (Similar differences between yortex lines ; in Other words, a
theory and experiment are also found ,. .,. , '

on tapered wings.) discrete trailing vortex of some

strength exists near the trailing

edge in this region (cf. Article 166). Such a departure from theory
was anticipated on theoretical grounds in Article 175. Wings can
be designed to eliminate this feature at cruising or climbing
incidence, but the form of wing-tip required is less easy to construct
and with most wings the feature is present ; it has been found at
full scale, and the close-up vortices must occasionally be taken into
account in connection with the controls of aircraft. The peaks
of pressure reduction are situated on the rear part of the upper
surface, so that they lead to a large increase in drag over small
areas, as described in Chapter VII. The theoretical induced drags
for given shapes are ideal minima, and in actual wings apply only
to part of the lift ; the remaining part would appear to have a
drag appropriate to uniform lift.

Effects of the foregoing and other discrepancies are minimised in
the principal use that is made of the theory in design, viz. to calculate
relatively small differences between wings of much the same type,
as illustrated in Article 177. The feasibility of this use rests upon
experiment. Observations of lift and total drag coefficients are
obtained on a series of aerofoils based on the same section, but of
widely different aspect ratios. Plotting CL against CD for each
results in a series of very dissimilar curves, as will be appreciated.

VIII] WING THEORY 327

When, however, the reduction formulae are used to correct every
point on all the curves to some common aspect ratio, chosen as
standard, the originally divergent observations are found, within
certain limits, to agree with one another substantially, and all to lie
within a narrow band through which a single curve may be drawn for
practical purposes. The limitations are, firstly, that the aspect ratio
must be greater than a minimum depending upon section and
incidence ; the minimum varies between 2 and 4. Secondly,
incidence must in any case be restricted ; the check has been success-
ful in some cases up to 15°, but the region of maximum lift should be
avoided.

BIPLANE WINGS

181. The two wings of a biplane are variously arranged as illus-
trated in Fig. 134. Distance between the planes is called gap (see

NEGATIVE STAGGER

FIG. 134.— THE LEFT-HAND FIGURE ILLUSTRATES POSITIVE STAGGER, h0 BEING
THE GEOMETRIC GAP.

If A, B are the centres of pressure, h is the Aerodynamic gap and <£ +he
Aerodynamic stagger. If, alternatively, A and B are fixed points, usually
located at one-quarter of the chords from the leading edges, <f>0 is the geometric
stagger.

note to figure). When the upper wing is immediately above the
lower wing at 0° and has the same dimensions, the arrangement is
called orthogonal. If the upper wing leads, there is said to be
positive stagger, the amount being expressed as an angle. In a
sesqui-plane the lower wing is of reduced span and usually of reduced
chord. With positive stagger the upper wing is occasionally set at
the greater incidence by 2°-3°, when the biplane is sai4 to have
ddcalage.

The two wings interfere with one another in various ways, and
even in an orthogonal biplane at 0° have different lifts. Thus, con-
sidering a region of the atmosphere traversed a little time previously
by a biplane, we expect to find two vortex pairs of unequal strengths.
The induced drags of the wings differ from one another and from
that of a similar monoplane.

328 AERODYNAMICS [Cfi.

Changes of Aerodynamic efficiency from one arrangement to
another and in comparison with the monoplane will be studied, but
it should be remembered that a particular layout may be adopted
for ease of construction, restrictions on span, pilot's view, and
similar practical considerations. Thinner wing sections can be used
for biplanes, partly on account of decrease in span for a given lift, but
more importantly by virtue of the external bracing so readily intro-
duced. The drag and weight of the interplane bracing tend to offset
this advantage, however, and also to limit the amount of stagger and
gap that can usefully be employed.

182. Some General Theorems

As for the monoplane, the trailing vortex lines behind each member
will, for purposes of calculation at the wings, be assumed to extend
downstream parallel to the direction of motion.

A slight extension of Articles 173-4 leads to the result that mini-
mum drag occurs for a given lift when both wings are elliptically
loaded and the induced velocity is the same at each. Elliptic lift
distribution will be assumed.

Introduce stagger, either positive or negative, at the same time
modifying the incidence of every section of either plane in such a
way as to ensure that the distribution of lift throughout the system,
whatever it may be, remains unchanged. By considering a region
of the atmosphere lately traversed, the kinetic energy generated
within it by the biplane is clearly seen to be independent of the
degree of stagger. Thus the total induced drag is, with the proviso
stated, unchanged by stagger. It does not follow that the induced
drag of either plane is unaffected, in fact the contrary will be found.
This important theorem is known as Hunk's equivalence, or stagger,
theorem. It enables a staggered biplane to be replaced, for purposes
of investigation, by an equivalent system of zero stagger, having the
same total lift and drag, but with drag distributed between the
planes in accordance with the degree of stagger.

Distinguish the upper wing by suffix 1 and the lower by suffix 2f
and denote an effect on 1 by its own vortex system by the suffix 11,
an effect on 1 due to the vortex system of 2 by the suffix 12, and so
on. The total induced drag of the biplane is —

An = An + A*a + Aii + Aii • • (246)
The first two terms on the right-hand side are calculated as for
separate monoplanes ; the last two represent the effects of inter-
ference. Regarding mutual effect, the circumstances of either wing

VHI] WING THEORY 329

are modified by (a) the bound, and (b) the free vortex system of the
other. We investigate the mutual effect as for (6) only, and intro-
duce a correction for (a) later.

Divide the span of each wing of a biplane of zero stagger into a
number of elements having equal small lifts. Each separate element
will produce trailing vortices, appropriate to its span, which,
though unequal in spacing and strength, induce equal velocities at
distant corresponding points owing to the equality of the lifts.
Consider a single pair of elements, one on each wing. If w' repre-
sents the induced velocity at the one element due to the trailing
vorticity behind the other —

w" XT w" XT
— 8L1== — 8I2.

This is true of any pair of elements. Hence, if AAn be the whole
induced drag of a chosen element of wing 1 as due not to other
elements of the same wing but to the effects of all the elements of
wing 2, and ADt21 be that part of the induced drag of the whole of
wing 2 due to the effects on all its elements of the single chosen
element of wing 1 —

AZ),lt = AA,I.
Therefore, by summation for all elements —

Ai2=A» .... (247)
Substituting in (246) we find —

AB = An + Ai. + 2A». - . (248)

It is important to remember that this result is for zero stagger.
With the aid of Hunk's equivalence theorem it becomes of general
utility, for by this we can replace a staggered biplane by a particular
unstaggered system, to which the above result may be applied.

Now, since elliptic loading is assumed, the induced velocity at any
point some distance downstream due to wing 2, say, can be calculated
on the analogy between its vortex sheet and a flat plate in broadside-
on motion by the methods of Article 117 or 124. The point can be
moved upstream to the vertical plane containing both wings by
introducing the factor ^. Writing 2$ for the span as before, we can
then calculate for zero stagger —

1 f *

An =Fr WiJLl9 . . . (249)
UJ -*,

and (248) can finally be evaluated with the help of the monoplane
formulae.

330 AERODYNAMICS

183. Prandtl writes (249) in the form —

[CH.

. (250)

where a depends on Sj/s,, and the ratio of gap to mean span. Fig.
135 gives his values for a through a useful range.

O4 05

FIG. 136. — PRANDTL'S FACTOR FOR BIPLANES.

Denote the quantity lift/span for each wing by \i or X,, so that
(250) takes the form—

VHI] WING THEORY 331

Since for a monoplane 2/nA = S/2ns*, the induced drag of a mono-
plane with elliptic loading is obtained in the same terms as —

• • <261>

Hence the induced drag of the complete biplane is found from
(248) to be given by—

t). . . (252)

This is a minimum for a given total lift when —

A j o j QTo ^

The value of the minimum is —

(2H)

where the monoplane drag is for equal lift, and a span equal to that
of the longer wing (1) of the biplane. The factor to be applied to
DiM is always < 1, so that a biplane has less drag than a monoplane
of equal lift and span. This result neglects, of course, the parasitic
drag of inter-plane bracing.

In the biplane of minimum induced drag and zero stagger, the
mutually induced drag is divided equally between the wings, and
we find —

(£/A)i = Si/(Xi + aX,) =

(LJD,), si/(X, + aX,)

on substituting for AJ or X, from (253), i.e. the wings have equal
lift/drag ratios. But this is not true if the total lift is differently
divided between them.

The disposition of lift required for minimum drag is usually dis-
advantageous from a structural point of view, since the longer wing
is much the more heavily loaded. It is therefore useful to note that
for practical variations the above minimum is not critical but ' flat/

184. Examples

To illustrate the significance of the foregoing results, we consider a
biplane lifting 2700 Ib. at 150 ft. per sec. whose upper and lower
wings are 30 ft. and 24 ft. span, respectively, with a gap of 5 ft.

332 AERODYNAMICS [CH.

For a monoplane of the same lift and 30 ft. span —

_ 2 x 8100 M „ „
D == - = 96-3 Ib.
' Tip X 22500

since XM == 90 Ib. per ft.

For 6-ft. gap <r = 0-48, approximately, and the minimum drag is —

96-3 (1 — 0-23) __
T~— 0-96 X 0-8 + 0-64 ^ 85*° '
We also have —

Xa = 15 - 5-76 = = L> X 24

X, 12 — 7-20 La X 30

whence Ll = 1907 Ib., La = 793 Ib. Since Xj = 63-6, Xa = 33-0 —

Dtl == 48 Ib., £>aa = 13 Ib., Dia + Dtl = 24 Ib.
and as the biplane has zero stagger,

total induced drag, upper wing = 60 Ib.
total induced drag, lower wing = 25 Ib.

We note a reason for diminishing the chord of the lower wing, but
that the difference in loading is rather excessive. It might be more
practical to make Xa = f Xlf so that Xj = 56-2, Xt = 42-2. We
should then have 86*9 Ib. for the total induced drag, made up as
follows :

Dn = 37-6 Ib., Daa == 21-2 Ib., Dia + D2l = 27-1 Ib.,

showing 13 per cent, increase in the drag due to interference, but
1 per cent, only in the whole. Since now Ll = 1687 Ib. and La =
1013 Ib., it is found that for zero stagger the lift/drag (induced) ratio
of the upper wing would be increased by 4 per cent, and that of the
lower decreased by 8 per cent.

The loss associated with the new loading could be recovered by a
small increase of gap. The only drag then affected is that mutually
induced. It is easily found that, to decrease this by the small
required amount, cr must be reduced to the value corresponding to a
gap of 5 ft. 4 in.

185. Equal Wing Biplane — Comparison with Monoplane

Article 183 shows that minimum induced drag occurs for a given
lift when the wings have equal span and X is the same for both,
i.e. their lifts are equal. We then have for the complete system —

I + CT)' • • • (255)

where XB is the total lift per unit span of the biplane.

WING THEORY 333

An approximate expression for 1 + cr when Si = sa = s is, from
Prandtl, writing h for the gap—

i L^ 2-065 + 1-52 (h/s)

1 -f- G = /LJ'~\ * • - (^5oj

The span-grading of a monoplane carrying the same load as a
biplane and having the same induced drag is immediately obtained
as —

AM «xB/Y/i-i-*. . . . (257)

Example. — If the gap is one-sixth of the span of the biplane, the
span-grading of lift must be reduced for the monoplane by the
factor 0-875.

The results may be expressed in terms of aspect ratio as follows,
the aspect ratio of the biplane being defined by AB = 8s* JS, where S
is the total area, so that for wings of the same dimensions As equals
the aspect ratio of either. From —

Hence —

CD* = -*r CLB (1 + <r) . . - (268)

TC/IB

For the same lift coefficient and aspect ratio the induced drag
coefficient of a biplane is greater than that of a monoplane in the ratio
(256). The lifts are then different, however, for equal span. To
make the lifts the same, the area of the monoplane must be doubled,
whence we find that the drag of the biplane, supporting the same load
at the same lift coefficient and having the same aspect ratio, is greater
than that of the monoplane in the ratio (1 + a)/l. It will be noted
that the monoplane span-grading of lift is then 1/V^ times that of
the biplane. Let us determine the aspect ratio of the monoplane
which will lead to the same induced drag at the same lift and lift
coefficient. Evidently we must have—

^M^^L.

A* 1 + a"

Example. For a gap-span ratio of 1/6, AM = 0-663 AB. This can
be written, since the areas are equal, 4s£ = 0-653 x 8s £, whence

334 AERODYNAMICS [CH.

2sM = 1-143 x 2sB. The chord of the monoplane is the greater by
75 per cent.

Incidence. — For biplane wings to achieve a given lift coefficient
their incidence (a) must be increased from that appropriate to their
sections under two-dimensional conditions (oc0) to a greater extent
than monoplane wings of the same section and aspect ratio.

On account of the double trailing vortex system a is evidently
increased to —

a - a0 + — CL (I + a) . . . (259)

A further increase is required on account of the curvature of the
streamlines in which each wing operates, due to the bound vortex
system of the other. This development is left to further reading.*
An approximate formula is —

0-025
Aa ----- --- c < (260)

(A X gap/span)2 L v '

Example. — For A = 6, a — a0 (in radians) for a monoplane with
elliptic loading = 0-053 CL ; for an equal wing biplane of gap-span
ratio 1/6 it is 0-082 CL at the lower estimate (259), which requires
increasing by some 30 per cent, to take account of the neglected
factor (260).

1 86. General Remarks

The form of (260) shows that the secondary, but important,
incidence increase depends essentially upon the square of the ratio
of the chord to the gap. Thus, however A is increased, the slope of
the lift curve of a biplane is considerably reduced from the two-
dimensional value. Formula (259) can be amended on the lines of
(245) for lift distributions other than elliptic. Applying the same
reasoning as to mutual effect to biplanes which, as is usual, have
positive stagger, shows that the forward wing should have the smaller
geometrical incidence. But the reverse is sometimes adopted, and
improves the shape of the lift curve past the stall. If both wings of a
staggered biplane are of equal span and carry an equal load, the
forward wing is easily shown to have less induced drag than the
other.

The comparisons which have been made with the monoplane
neglect increase of form drag of the biplane wings due to their

* Bose and Prandtl, Zeits. f. ang. Math. u. Mech., vii, 1927.

VIII] WING THEORY 335

greater incidence for a given lift coefficient. They also neglect the
experimental value of the maximum lift coefficient, which is the
lower for the biplane and affects choice between the two in practice.

WIND-TUNNEL CORRECTIONS ON AEROFOIL TESTS

187. Enclosed Tunnel Constraint at the Aerofoil

When an aerofoil is tested in a wind tunnel of the kind in which the
stream is enclosed within walls, the walls diminish the induced
velocity at the aerofoil, which consequently experiences a fictitious
reduction of induced drag and incidence. If the aerofoil has the
same aspect ratio as the wing it represents, observations of drag and
incidence must be suitably increased to apply to free air conditions.
This course of correction is that usually followed. Alternatively, the
measurements might be made on a model of appropriately smaller
aspect ratio. The advantage of the latter method is that the aero-
foil may have a larger chord, and is then easier to make accurately
for small tunnels.

The constraint is often calculated with sufficient accuracy by the
approximation mentioned at the end of Article 166, replacing the
trailing vortex sheet by a vortex pair, and the actual tunnel wall by a
circular one. The distance apart / of these vortices is determined
from the lift of the aerofoil. For example —

if elliptic loading is assumed. Whence in this case —

'=? ..... (261)

Let the radius, or effective radius, of the tunnel be a. Assuming
the aerofoil to be located centrally, the images are distant a*\\l
from the centre. The upward velocity at the centre due to these

- = . =

4?c tf1 4C '
where C is the cross-sectional area of the tunnel. But

or —

IK. = |SC7CL

336 AERODYNAMICS [CH.

if CL is the lift coefficient and S the area of the aerofoil. Substituting —

1 S

-W----CLU.

(262)

This result has been obtained without use of (261) and applies to
any aerofoil, i.e. assumption of elliptic loading is unnecessary in
the present connection.

It is usually sufficiently accurate to increase w uniformly along
the span by this amount to obtain free air conditions. We then
have finally —

- l~C
Similarly— I ... (263)

1-0

T
oz

for a tunnel of circular sec-
tion. It will be noticed that
the distance / has vanished,
and that the corrections are
proportional to lift only.
Thus they apply even to a
biplane or triplane model,
when S becomes the sum of
the aerofoil areas.

Expressions of the same
form are obtained by the
method of Article 153 for
tunnels of square or rect-
angular section, the nu-
merical factor 0-125 for the
circular section alone being
4 changed. Fig. 136 shows
the variation of this factor,

FIG. 136. which is denoted by \T9 for

rectangular tunnels. It will

be seen that the variation from 0-125 is only 10 per cent, for open
sections.* The whole correction is usually less than 10 per cent.,
when this variation comes to less than 1 per cent, of the final
estimates- But where the clearance between the aerofoil tips and
the tunnel walls becomes less than one-fifth of the span, the

* Terazawa, Tokyo Repts. 44, 1928.

1 H,
H/B

WING THEORY

337

ao-

VIII]

foregoing approximate
method begins to be insuffi-
cient and the form of the
distribution of wf to have an
appreciable effect.

Examples. — The upper
curve of Fig. 137 would be
expected from an aerofoil of
R.A.F. 38 section, of 4-in.
chord and 24-in. span, at a
speed of 150 ft. per sec. in
a closed-section wind tunnel
of 4-ft. diameter. T . S/C =
0*0133, and multiplying by
half the square of lift coeffi-
cients gives increments of
drag coefficient leading to the lower curve for free air conditions.
Incidence would also be increased for a given CL, e.g. at CL = 1-0,
Aa= 0-0066 radian =0-38°.

What aspect ratio would a model of the same chord require for the
two curves to coincide ? Comparing (263) with (235), we have, dis-
tinguishing free air conditions by the accent —

06 08

FIG. 137.

is_i/a _JA

8C~7cU A')9

or, since C = TOI* and A' = 6 —

!_ — -i

~A 8a» ~ 6*

By (230) S = 4s2/-4, while 4sa = A*c*t c denoting the constant chord.
Hence, or more directly —

1 Ac* _ I

A ~~ *8o» ~" 6*

Substituting a = 2, c = 1/3, gives A = 5-4. Thus an aerofoil of
4-in. chord and 21-6-in. span in a circular-section tunnel of 4-ft.
diameter would give through a limited range of incidence the same
lift and drag coefficients as an aerofoil of the same section but aspect
ratio 6 at the same Reynolds number in free air.

More exact conversion formulae may be developed to take account
of the actual lift distribution of the model tested. But the changes
following this refinement are of the order of 10 per cent, in practice,

338 AERODYNAMICS [CH.

representing, as already mentioned, a final variation of usually less
than 1 per cent.

188. Open-jet Tunnel

When aerofoils are tested in a free jet, corrections are required to
allow for its limited section. These are obtained from the appropri-
ate image system, which differs essentially, however, from that for an
enclosed tunnel of the same section. Whereas in the latter case the
criterion determining the image system is cancellation of velocity
components normal to the walls, with the open jet it is that the
pressure at the surface of the jet shall be constant and equal to the
pressure of the surrounding air at a distance. The new requirement
entails that the tangential velocity at the surface of the jet be reduced
to its value in the absence of the aerofoil. Thus tangential, instead
of normal, velocity components due to the aerofoil are to be cancelled
by the image system.

To verify this, let p0 be the pressure and U the velocity just within
the jet before introducing the model, which adds small increments of
velocity u, v, w there. By Bernoulli's equation —

Hence —

po — p = $>Uu,

neglecting squares of small quantities, so that for the pressure to
remain the same as outside the jet, u must vanish.

An approximate solution is easily seen in simple cases. Take first
the case of a two-dimensional aerofoil situated near a parallel flat
fluid surface of infinite extent, beyond which the air is at rest.
Locate the image as for a wall, i.e. at an equal distance beyond the
surface, but reverse the sign of the image, so that circulation round it
is in the same direction as round the aerofoil and the two form a
biplane of zero stagger. The tangential velocity component due to
the combination evidently, from symmetry, vanishes at the surface.
The normal velocity component there is doubled, so that the surface
is slightly bent, but this effect is often neglected.

The conformal transformation of Article 155 may be applied to a
two-dimensional aerofoil in a two-dimensional jet, reversing the sign
of the single image in the transformed plane, but not easily.

Take next the important case of a vortex pair symmetrically
situated in a jet of circular section, radius a. Reversing the sign of
the images at the inverse points gives the system of Fig. 138. With

vni]

WING THEORY

339

FIG. 138.

the notation of the figure it is easily verified that the tangential
velocity at the general point P is proportional to —

(B — s B + s

sin v I ----- - ------ ---

V fi2 ra2

B + a*/s B
].--L j

r32

a*/<

The expression within the brackets vanishes. Thus the artifice of
changing the signs of the reflected vortices again succeeds in regard
to the tangential velocity, but the normal component is again varied.

In practice, the correction formulae of the preceding article are
applied to an open jet with their signs changed. But the step is
tentative and rests upon experimental justification.

This method is also used for jets of elongated section, such as
elliptic or rectangular. For general treatment reference should be
made to a paper by Glauert.* A simple rule appears from this and
other investigations : — The correction formulae for enclosed tunnels
apply to open jets of the same sections, provided the sign is changed,
and also that the aerofoil is rotated through a right angle.

If, with the last proviso, a small aerofoil is tested in an open-jet
tunnel and also in an enclosed tunnel of the same size and section,
the mean results should give the free air coefficients and incidence.

It must be confessed, however, that the theory of the correction
for constraint, being based on ignoring the distortion of jets, is not
well founded in their case.

DOWNWASH AT TAIL PLANE

189. The tail plane of an aeroplane is required to exert, with eleva-
tors neutral, zero pitching moment about the C.G. of the craft at some

* A.R.C.R. & M., No. 1470.

340 AERODYNAMICS [CH.

arranged speed and wing lift coefficient (Article 88) . This is achieved
by a suitable tail-setting angle. Tail planes may be adjusted (or
their lift, e.g. by trimming tabs), but it is desirable to form a close
estimate at the design stage of the required tail-setting angle, which
depends upon the downwash at the tail position (Article 86).

Unfortunately, the magnitude of the downwash affecting the tail
plane is difficult to calculate owing to the lack of precise know-
ledge of the trailing vortex configuration at this intermediate position
behind the wing. The vortex sheet will have partly, but not
completely, rolled up, while the existence of any wing-tip vortices
close to the wing will affect the calculations.

A rough estimate is obtained by substituting for the actual wing a
hypothetical one of equal lift distributed uniformly along a suitably
reduced span 2s' (cf. Article 166). We then easily find that at the
level of the aerofoil and at a distance x downstream from its C.P.
the downwash angle is given by —

- K \
47cC7

The first term in the curly brackets is the contribution from the
wing, the remainder that from the fully developed vortex pair, while
K is the uniform circulation of the simplified vortex system. The
expression reduces to —

XL. A/(s'' + *«n , v

(264)

This result is readily expressed in more practical terms. Let the
actual wing be of span 2s and aspect ratio A and let CL be its lift
coefficient and a its incidence.

For uniform loading $' = 5, and the factor K/2nUs = CJZnA.

Assume alternatively elliptic loading along 2s. We then have
for the factor —

~ n*A ' s' L

from (231), while from (226) s/s' = 4/w, giving for the factor
Then using (236)—

Put, for example, % = s to represent a possible position of the tail
plane. Then we have for that position —

VIII] WING THEORY 341

giving for 4=6, for instance, de/da = 0-46. But it should be
remembered that such values assume a lift coefficient slope of 2n in
two-dimensional flow. Results for other lift distributions along the
actual wing are obtained in a similar way.

However, for the reasons stated, (264) cannot be regarded as
adequate, and it is more reliable to determine e by model experiment
(Article 86) . Observations of down wash require correction for wind-
tunnel constraint, and the same difficulty arises in determining the
amount of this. On the other hand, we then calculate only a small
correction, and error is of far less significance.

190. Tunnel Constraint at Tail Plane

When a complete model of an aeroplane is tested in a wind tunnel,
the downwash at the tail plane differs from that in free air. Tail-
setting angles observed in an enclosed-type tunnel must be increased,
and those observed in an open jet reduced, to allow for the limited
expanse of the stream.

As in the preceding article, substitute for the aerofoil one of appro-
priately shortened span and uniform lift with a fully developed vortex
pair springing from the wing-tips. Assume this to be arranged
symmetrically in the tunnel and restrict attention to the constraining
velocity w along the tunnel axis at distance x behind the C.P. of the
aerofoil through which the bound vortex lines are supposed to be
concentrated. The constraining velocity WQ> say, at the aerofoil has
already received discussion, while its value w^ far downstream ==
2w0 (Article 148) ; the present problem is to determine intermediate
values.

Even with the simplifications adopted analysis tends to be com-
plicated. Glauert and Hartshorn * have obtained :

0-24 CL . . (266)

for an enclosed tunnel of square section, of side H and area of cross-
section C, S being the area of the aerofoil and CL its lift coefficient.
In the above form the formula may be applied also to biplane
models, and modifications of the numbers may be introduced for
sections other than square. The formula is especially arranged to
hold up to distances downstream representative of normal tail-plane
positions, but, the approximation being linear, it must not be applied
to greater values of x. Kdrmdn and Burgers f have calculated by

* A.R.C.R. & M., 947, 1924. f Aerodynamic Theory, ii, 1935.

342 AERODYNAMICS [CH.

means of Bessel functions the constraint in open and enclosed
tunnels of circular section. Instead of reproducing these investiga-
tions, we shall estimate in an approximate way the constraint for a
circular section, using the methods of Chapter VII.

191. Estimate for Circular Section

Let a be the radius of the jet or enclosed tunnel, 2s' the span of
the equivalent aerofoil of uniform lift, K the circulation round the
simplified vortex system, wv the velocity at x due to the images of the

FIG. 139.

vortex pair, wa the velocity at % due to the image system of the
circulation round the aerofoil.

For the total constraining velocity w at x, which is to be subtracted
from observation in a jet or added to that in a walled tunnel, we have,
omitting sign, w = wv + wa. It is convenient to express velocity
contributions in terms of w«>. Thus —

(I)

W/WK = J at the aerofoil and = 1 far downstream, where wa
vanishes.

Let the distance of the image of each trailing vortex from the axis,
which passes through the centre of span, be^. Then^y = af/s' and —

* K K*'
= 2—=—. . . (u)

rca*

From Fig. 139—

VIII] WING THEORY 343

and therefore —

^=1(1+ cosY) • • (iii)

where cos Y = %IV(X* + y1)-

Turning to wa, substitute for the actual circular boundary a square
contour of side H enclosing
an equal area, so that — .+ .+ +.. .+ 4

H* = Tea1 or H = ai/n (iv)

The image system of the «-— . — — ^— ....—

aerofoil AB (Fig. 139) is
indicated in Fig. 140 for

an enclosed tunnel (cf. .4- + 4- 4-
Article 153), both columns
and rows extending in-
finitely.

If the images in the
rows were continuous and
two-dimensional conditions

held, the velocity at x FIG. HO.

due to the real vortex
would be decreased by the walls in the ratio (cf. Article 155)—

sinh (nxjH)'

Since an image of length 2s' occupies each length H of the rows, we
assume as an approximation that this constraint is to be reduced by
the factor 2s' /H.

The velocity at x due to AB in free air is (Fig. 139)—

— - 2 cos p

47T*

where cos p = s'/Vfc'1 + x2).

Applying the approximation gives —

K n

^^^^

and since w* = Ks'/H* —

- _ i

nxHJ

-nx/H sinh (nx/H).
Hence, using (iv) and by (i) and (iii) —

-^ = j| 1 + cos Y + 2 cos p f— ~ * 1 1. (267)

w« I Lx^n/a sinh (xi/n/a) J J v ;

344

0-8

Woo

0-6
±0-5
-0-6
-0-7
-0-8

-A.Q

AERO

s'.s,

a 4>

S1 i
a z

2 0-

4 ,0-

x/a

6 0

8 1-

v O

^2,

S'.l
?2

S'S^

a' 4"

~-t

FIG. 141. — CONSTRAINT WITH A
CIRCULAR STREAM.

The circles represent Karman and
Burgers' results, the curves the approxi-
mation (267). The upper curves show
corrections to be added to velocities
observed in an enclosed tunnel, the
lower curves those to be subtracted from
observations in an open jet.

NAMICS [CH.

Kdrmdn and Burgers' results
are given as suitable for s'/# n°t
much exceeding £ ; they are
shown as circles in Fig. 141, the
upper half of the figure applying
to an enclosed stream and the
lower to a jet. The curves are
obtained from (267) with s'/# = \
and f . The curves in the upper
half of the figure for an enclosed
stream are reflected in the #-axis
to apply to a jet.

A model tail plane will seldom
be farther downstream than \a
in an enclosed tunnel or \a in an
open jet. Thus (267) appears to
give a good approximation.

1 91 A. Application

Referring to a monoplane in
free flight, let e be the angle of
downwash in the neighbourhood
of the tail plane and a the inci-
dence of the wings. A usual problem is to determine rfe/^a from an
estimate of rf£0/rfa0, the corresponding quantity measured in a wind-
tunnel experiment with a model. Let the working section of the
tunnel be enclosed and have a cross-sectional area C, and let the
area of the aerofoil be S. Then (ii) of the preceding article can be
expressed in the form —

^ _ 5
~U ~~ * C Ll

as may be written down alternatively from (262). Denote by B
the R.H.S. of (267), which will be known from the conditions of the
experiment. Then —

e =

and —

de
fa

Wan + 4C

don

(i)

VIII] WING THEORY 345

This expression gives a close estimate of de/da provided the slope
of the lift curve of the model in the tunnel is also measured. For
by (263)—

8C 8C rfa '

and substitution for dC^/da. can be made without much resulting
error from (236) or a modification of that formula.
Alternatively, (i) can evidently be written —

ds dsQ d&Q BS rfCL ....

dy. d(x,Q da, 4C da. '

a suitable form if the slope of the lift curve is known accurately for
the monoplane in free flight.

192. Tail Planes of Biplanes

Superposition of monoplane results to obtain those for multiplane
wing systems has already been discussed in Article 86. It must be
remembered, however, that dCLjdu. is less for a biplane than for one
of its wings separated as a monoplane. Accordingly, de/da. is less
than double its value for the monoplane. The factor 0-8 may be
applied for aspect ratios in the neighbourhood of 6.

Chapter IX
VISCOUS FLOW AND SKIN DRAG

193. In Chapters V-VIII the viscosity of air was ignored, except
in accounting for the production of vorticityin the simplified distribu-
tions assumed. Neglect was justified by the successful calculation of
practical velocity fields, surface distributions of pressure for slim
shapes, the lift and induced drag of wings, and other results of
common utility. The artificiality of infinitely thin boundary layers
(Article 43) prevented any investigation of skin friction. With
modern aircraft of large speed range, however, owing to elimina-
tion of form drag and the small lift coefficients normally in use, skin
friction is of paramount importance.

Attention is now turned to the force arising within the boundary
layer as distinct from that due to pressures transmitted through it,
although these two forces are not, of course, independent of one
another. If the surface of the body is Aerodynamically smooth in a
sense that will be explained later, the force arising is a pure skin
friction, as introduced in Chapter II. But the slight roughness
of surface of many aircraft bodies is not negligible, and
introduces additional drag of the nature of a finely divided form
drag. The two components together constitute skin drag. For
the present we assume sufficient smoothness to avoid the second
component.

On reinstating viscosity, calculation immediately becomes diffi-
cult, and a feature of our new study is that analysis alone cannot go
very far. Mathematical complexity arises essentially from the fact
that the flow within long boundary layers at aircraft speeds is for the
most part turbulent (Article 21). On the other hand, corresponding
boundary layers of experiment may be largely steady. Again,
although the whole friction of a body can be measured with com-
parative ease, the determination of its distribution even round a
model in a wind tunnel is by no means simple.

These and other difficulties necessitate oblique attack from several
angles, and some of the problems studied are selected for convenience
and simplicity rather than on account of their direct application to

346

CH. IX] VISCOUS FLOW AND SKIN DRAG 347

aircraft. Such application demands greater intuition and empiricism
than is usually called for in Aerodynamics.

PIPE FLOW

194. Parallel, i.e. strictly laminar, flow obtains nowhere past an
aircraft, neither does the special type of turbulent flow occurring in
pipes at large Reynolds numbers, which is constrained by the long
parallel wall to a uniform profile of time-average velocity. The
subject is of interest, however, partly as an introduction and also in
view of a practical use to be deduced by semi-empirical reasoning.

In experiments with long pipes, fluid is commonly supplied to the
mouth or inlet in an agitated state. Initial disturbances usually
develop along the pipe into turbulent flow, but in some circumstances
they are damped out. The run of pipe required to achieve damping,
when this is possible, is called the stilling length. Flow for some
distance from the inlet is the same as that in an enclosed-type wind
tunnel ; a boundary layer lines the wall, and, increasing in thickness
along the pipe, accelerates by its obstruction a central stream whose
pressure diminishes according to Bernoulli's equation. After a
1 transition length ' the boundary layer fills the whole of the section
and, assuming damping, laminar flow becomes established. The form
of the laminar flow can be calculated by a development of the method
of Article 24.

Steady Flow between two Fixed Parallel Plates.— This is the
simplest problem after that of uniform rate of shearing (Article 24).
The plates are supposed so
large compared with their —
distance apart h that edge
effects may be neglected. —
Draw Ox (Fig. 142) in the ~
direction of motion midway
between the plates, and Oy
perpendicular to them. If
the flow is steady, the
streamlines are every- —
where parallel to Ox ; there FIG. 142.

is no variation of the pres-
sure p except in the direction Ox, and this variation is a constant
gradient ; i.e. —

OA ^ j*. oj%

{j-p ofi Cj)

-i~ x= ^- = 0, ~- = a constant = P, say.
oy dz dx

348 AERODYNAMICS [CH.

Consider unit length and depth of a stratum of fluid parallel to the
plates of thickness Sy. In the direction Ox the traction on the lower

_ A ^S

face is — \i — , while that on the upper face is ^ — (u + — 8y), or

32w
there is a resultant traction LL — - 8y. The force exerted in this

9ys

direction is — P . Hy. Hence, since the motion is steady —

da« ^
|i — - P = 0.

r dy*

On integration —

u = fy* + Ay + B . . (i)

Z\L

The condition of no slip at the boundaries (Article 22) states that
u = 0 when y = ± i^> giving two equations for determining the
constants of integration A and B :

From these 4 = 0, B = — PAa/8|ji. Substituting in (i)—

u = — (4y« - A'). . . . (268)
8{j.

The distribution of velocity is parabolic, as shown in the figure.
For its mean value u we have —

1 f*/3 P

U = T W^V == — rr- A1. . (U)

AJ -A/2 J 12 p, v ;

The propulsive force on the whole mass of fluid per unit length and
breadth of the plates is — PA, and must be balanced by the traction
on the two plates. Hence, if T * is the intensity of skin friction on
either plate, — 2r — PA = 0 or —

T = -iPA . . . (iii)
Alternatively, we can calculate T from the formula (cf . Article 24) —

obtaining the same result if in this case we draw y from the

* It was not possible to use this symbol for the friction per unit area in Chapter II,
but the change is now made to a nomenclature which is international. Suffix 0
for the boundary value is omitted where no misconception can arise.

IX] VISCOUS FLOW AND SKIN DRAG 349

surface concerned into the fluid. The friction coefficient of either
plate is —

pfi« 2p

where the Reynolds number R = wA/v.
The vorticity £ (Article 39) reduces to —

„ du Py

< — * — 1? ' ' (1V)

being zero along Ox, but, away from the axis, having values propor-
tional to distance from it, rising to the maxima i %Ph/y. at the
plates. If the channel formed between the plates be supposed fed
with fluid in an irrotational state, vorticity is seen to be generated by
the action of the boundaries and viscosity.

The above results should be compared with those of Article 24.
For the same coefficient associated with the uniform rate of shearing
there examined, we have, since u = \U —

£-£•"5-1 ' ' ' (270)

defining R in the same way as for (269). If, on the other hand, the
velocity U of the moving plate were selected to specify R, which is a
matter of choice, the friction coefficient would be l/R.

195. Steady Flow through Straight Pipe of Circular Section

The pipe is supposed to be very long and only a central length is
considered. The assumption of steadiness clearly means that the
pressure is constant over each section of the pipe ; its gradient in the
direction of flow (Ox) is an absolute constant, P. Consider unit
length of a thin concentric cylindrical shell of internal and external
radii r and r + Sr. The propulsive force on the shell due to the
pressure gradient is — P . 2nr . 8r. In the direction Ox the resultant
of the internal and external tractions comes to —

Therefore, since the flow is steady —
Integrating —

u==-~r* + Alogr + B . . (i)

350 AERODYNAMICS . [CH.

Along the axis of the pipe, where r =• 0, u cannot be infinite, so
A = 0. Denote the bore of the pipe by D. When Y *= \D the
boundary condition of no slip states that u = 0, whence (i) gives

PD1

Hence the expression for the velocity reduces to —

0«) . . . . (271)

The velocity profile is a paraboloid, the speed at the centre being
twice the mean, which is given by —

4 D/3

The propulsive force on the whole mass filling the pipe is — - PD1

per unit length, and equals the retarding traction 7 . nD at the wall.
Hence, if R = wD/v, the friction coefficient is —

^=1 • • • • <272>

196. Comparison with Experiment

The laws demonstrated in the preceding article were first found to
hold for small Reynolds numbers nearly a century ago by Poiseuille
and Hagen experimenting independently. This early success fur-
nished a valuable proof of the conception of zero slip at the
boundary. Some fifty years later Reynolds established that if the
fluid at inlet is in a disturbed state, laminar flow can only result
when the Reynolds number (uD/v) is less than 2300, approximately.
Experimenting with water in a glass pipe, he showed that a little
colouring liquid introduced at inlet formed, below the ' critical
Reynolds number/ a steady line parallel to the axis. At greater
scales the colouring matter could not be followed, becoming mixed
with the stream, which developed a turbulent motion. Later on,
Couette examined the jet of water issuing from the outlet end of a
pipe. Well below the critical Reynolds number it was crystal clear ;
well above, it presented a frosted appearance, whilst at the critical
stage it oscillated between these two states in a periodic manner.*
Simultaneous changes in the trajectory of the jet showed a greater
resistance of the pipe to turbulent than to steady flow.

By fitting a nipple at each end of a central length of a long pipe
and connecting to a pressure gauge, accurate measurements are easily

* Tietjens, Applied Hydro- and Aeromechanics, p 37.

IX]

VISCOUS FLOW AND SKIN DRAG

351

made of the resistance of the length through a wide range in rate of
flow, the latter being measured by weighing, with a liquid, or by
feeding the fluid through a calibrated orifice in the case of air. A
number of such investigations has been carried out with smooth
pipes, diameter and fluid being varied, and they provide an excellent
check on Rayleigh's formula (Article 47). The dots given in Fig.
143 are mean values obtained from the tests of Stanton and Pannell,

-2

-3

-4

(272T

,(273)

2-5

FIG. 143. — FRICTION IN STRAIGHT PIPES OF CIRCULAR SECTION WITH INITIAL

TURBULENCE.

Saph and Schoder, and others. The result (272) exactly fits these
readings as far R — 2000. The friction coefficient at the critical
Reynolds number is vague, but for the turbulent flow thereafter the
coefficient is increased greatly above its value for laminar flow.

Numerous investigators have demonstrated more recently that
if the disturbances at inlet be reduced below a certain small maxi-
mum, laminar flow can be established at still higher Reynolds
numbers, and by supplying the pipe with an extremely smooth flow
the critical stage has been advanced to R — 20,000.

Experiments of the above kind are frequently undertaken as
providing valuable laboratory work, and some precautions against
error may be noted. The pipe should be fitted with a bell-mouth
at inlet and a stilling length of at least 60 diameters allowed. It
should be constrained, if necessary, to straightness ; considerable
curvature of the axis produces a steady streamline flow of dissimilar
form, the centrifugal pressures introduced giving rise to a double
corkscrew motion, increasing resistance. Commercial tubes are
seldom round, but errors on this score are less important than those
due to taper, when kinetic energy must be progressively added to the

362 AERODYNAMICS [Cfi,

stream. A similar error is caused by an insufficient transition
length. The laminar velocity profile is approached only asymptotic-
ally, and meanwhile kinetic energy is added near the axis, increasing
pressure drop. Pipe flow is often employed to calibrate anemo-
meters intended for use very close to a boundary, e.g. the surface of
an aerofoil. In such cases it is important to approximate closely to
the calculated velocity profile, when a very generous transition
length is required. If large critical Reynolds numbers are desired,
great care must be exercised to free entering fluid from even such
small disturbances as convection currents. In some Aerodynamic
laboratories air will be used as the fluid. The orifice box surround-
ing the inlet should then be of ample proportions and the gauge
recording intake pressure difference should be calibrated for low
speeds by an aspirator method.

197. Turbulent Flow in Pipes — the Seventh-root Law

The unsteady flow beyond the critical Reynolds number is not
susceptible to calculation. When referring to the velocity at any
radius we mean the time-average value there. Sufficiently far from
the inlet the profile across the section of the time-average velocity
remains constant along the pipe ; it is much flatter than for laminar
flow, the maximum velocity at the axis being approximately 1-24
times the mean, and the gradient at the wall being steep.

The simplest empirical formula for the friction coefficient (Blasius)
is —

~ =0-0395 tf~1/4 . . . (273)

pw2 v '

and holds as far as jR = 10* as shown in Fig. 143. A more general
formula, due to Lees,* extends agreement to R = 5 x 106, but at
greater scales divergence again occurs.

On the basis of (273), if T denote the skin friction at the wall —
T = 0-0395 p«ajR~1/4 = 0-0395 pfi'/V'4/)-1'4 (i)

If we assume that the time-average velocity u distant y from the
wall can be related to the axial value um by the simple formula —

« = u» (yl*Y = l-24fi (y/a)» . (ii)
where a = JD, the radius of the pipe, and substitute in (i), we find —

T = 0-0228 pv1' V/4y ~ 7»/Vn/4 - 1/4 . (iii)

Now it may be assumed, as an approximation, that as u increases
the velocity profile retains its shape, when n in (ii) will be a constant.
* Proc. Roy. Soc., A, v. 91, 1015.

IX] VISCOUS FLOW AND SKIN DRAG 353

If we further assume that close to the wall u is independent of a, it
must follow that —

in - J = 0
and —

n = \ (274)

This law, due to Prandtl, is (rather surprisingly) found to hold
closely throughout the greater part of the section of the pipe, break-
ing down only near the axis and the wall. It is restricted, of course,
to the same range as (273), on which it depends, but at greater
Reynolds numbers the index may be progressively decreased as an
approximation ; at about 107 it becomes 0-10.

The great increase of resistance to turbulent as compared with
laminar flow shows that molecular motion can no longer account for
the transverse transport of momentum through the bulk of the
stream. Past the critical Reynolds number molar masses of fluid
penetrate from one radius to another and, compared with the
resulting tractional stresses in the fluid, laminar friction is small.
This is not the case near to the wall, however, where molar move-
ments must eventually cease. Thus the skin friction is ultimately
bu.lt up by the same mechanism as for steady flow. The thin film
adjacent to the wall through which viscosity predominates is known
as the laminar sub-layer, but observations with the ultra-microscope*
prevent its being regarded as in a steady state.

FIG. 144. — VELOCITY DISTRIBUTIONS FOR STREAMLINE AND TURBULENT FLOW

THROUGH PIPES.

Through this layer the seventh-root law evidently fails, for we
find on differentiating (ii) that T becomes infinite as y vanishes. To

* Fage and Townend, Proc. Roy. Soc,, A, v. 136, 1932.
A.D.— 12

354 AERODYNAMICS [CH.

avoid the anomaly we suppose the velocity profile drawn in accor-
dance with (274) to hold only from near the axis to the edge of the
laminar sub-layer, and to be joined to the wall by a straight line
having a slope dictated by the known value of T (Fig. 144).

PIPES WITH CORES
198. Annular Channel

Some problems of practical Aerodynamic interest are con-
veniently studied in a qualitative manner, both analytically and
experimentally, by considering flow through a long pipe fitted with a
core that extends through its entire length. The core may be so
small compared with a pipe of convenient diameter as to represent,
geometrically, a very narrow body in, for example, a 5-ft. wind
tunnel. In these circumstances it is possible to neglect, if required,
as an approximation, effects of the core on the resistance of the pipe
wall and of the general velocity profile on that of the core.

The solution for laminar flow through a circular pipe with a
concentric circular core is easily deduced from Article 195. If the
core is of diameter dt the constants in (i) of that article are now to be
determined with the additional boundary condition : u = 0 when
r = \d. To evaluate A and B we have —

and a similar equation with d written for Dt whence

and for the mean velocity —

4 fD/a P

Comparison with Article 195 (ii) shows that a central core of
diameter only a few thousandths of that of the pipe suffices to
decrease the flux for a given pressure gradient, i.e. to increase
resistance, considerably. With large cores the friction approximates
to that for flow between parallel planes.

According to some systematic experiments * the critical value of
«Z)/v at which turbulence develops is delayed 16 per cent, by
small cores, 50 per cent, where d/D = 0-15 — 0-5, and increasingly
for narrower annuli. But the range for parallel flow is actually

* Piercy, Hooper, and Winny, Phil. Mag.t Ser. 7, v. 15f 1633. (The subsequent
article is based on the same paper.)

VISCOUS FLOW AND SKIN DRAG

355

4000

V
2000

1000

O-2

0-4

d/D

FIG.

145.- CRITICAL REYNOLDS NUMBERS
FOR PIPES WITH CORES.

5 (The broken line gives an approximation
based on hydraulic mean depth.)

(By permission of the Phil. Mag.)

IX]

greatly reduced by small cores, a more or less periodic motion of
swaying type setting in, owing probably to small variable eccentri-
city, which increases resistance (such ' secondary motions ' often occur
in flow through other than
straight circular pipes, e.g. 5OOO
in curved pipes, and are not
to be confused with turbu-
lence). Thus upper and
lower critical speeds occur
as shown in Fig. 145. The
broken line in this figure
illustrates a basis of ap-
proximation that is often
used when, as in some ap-
plications to Aerodynamics,
it is sought to correlate
results for pipes and channels
of different sections. This
curve derives the critical
values of wD/v on the as-
sumption that they will

vary as the hydraulic mean depth, defined as the ratio of the cross-
section of a stream to its wetted perimeter. For annular sections
this ratio is evidently (D — d)j± ; for a pipe it is Z)/4 ; so that the
curve gives appropriately R'D/(D — d), where R' applies to the
pipe without a core.

Attempts have often been made to relate the incidence of turbu-
lence in various cases to a common value of the skin friction coefficient.
White * found that T/pw2 = 0-0045 applied approximately in this
connection to pipes of various curvatures. The value for the
straight pipe = 8/2000 = 0-004. For all annular channels this
coefficient lies between 0-0045 and 0-0074. According to the author's
experiments, turbulence is developed through wide annuli (djD <
0-5) at mean velocities — with a given fluid and pipe — that are pro-
portional to the ratio of the whole friction to that on the pipe wall
only.

199. Eccentric and Flat Cores

The analytical problem of eccentric cores and of cores of other than
circular section is a little complicated, and must be left to further
reading, but some results of interest will be described briefly.

* Proc. Ray. Soc.. A, v. 123, 1929.

356

AERODYNAMICS

[CH.

FIG. 146. — ISO-VELOCITY LINES FOR A PIPE WITH AN ECCENTRIC CORE.
(By permission of the Phil. Mag.)

Fig. 146 gives the velocity contours for steady flow in a typical
case of eccentricity. The flow is notably reduced through the
constricted side of the channel, its maximum velocity being only
30 per cent, of that on the open side. The resistance is 12 per cent,
less than with the core centrally situated. A similar obstruction to
flow through the passage between two bodies is often encountered in
Aerodynamic circumstances beyond means of calculation, and the
gap is filled in when small.

Fig. 147 shows the percentage increase of resistance to flow

through a long pipe, in which
there is a core of one-hundredth
part its diameter, for varying
eccentricity of the core. When
eccentricity is a maximum and
the core extends as a single
corrugation along the pipe wall,
the increase with steady flow is
negligible. Experiments show
the effect of cores to be less
marked in turbulent than in
streamline flow. In the absence
of direct experiments, the result

is reassuring as to the drag of a very shallow ridge, much less deep
than the boundary layer and parallel to the flow, introduced for
constructional reasons along a more or less flat surface.

In contrast, Fig. 148 indicates the theoretical laminar flow
friction variation across a flat core extending as a central strip

Eccentricity-i- Pipe

FIG. 147.

IX] VISCOUS FLOW AND SKIN DRAG 357

within an elliptic pipe. The
rapid increase in friction as the
sharp edges are approached will
be seen, and such edges along
the flow should clearly be
avoided.

Comparing the total friction
of a long flat strip with that
along the exterior of a circular
cylinder, they come to the same
in laminar flow if the diameter

of the cylinder is one-half the width of the plate. This result was
first obtained by Lees,* considering the resistance to motion of a
long plate through fluid contained in a wide stationary cylinder of
confocal elliptic section ; it obtains also for the corresponding case
of pipe flow, provided the cores are very small.

FIG. 148. — VARIATION OF FRICTION
ACROSS A THIN FLAT STRIP WITHIN
A PIPE OF CONFOCAL ELLIPTIC SEC-
TION (LAMINAR FLOW).

GENERAL EQUATIONS FOR STEADY VISCOUS FLOW
200. General Motion of the Element

When the flow is steady but non-laminar in the strict sense of the
word (Article 21), the element is subject to acceleration, although the
velocity at any point in the field is constant. As the element pro-
ceeds on its path, it is also subject to variation of vorticity and to a
certain stretching under the viscous stresses. We first reduce this
compound motion to the simplest terms necessary for framing
equations of motion. The matter is illustrated for steady two-
dimensional flow, parallel to the Ay-plane.

Let u, v be the velocity components parallel to Ox, Oy of the
centre G (xt y) of any small fluid element. The component velocities
at an adjacent point Q (x + 8x, y -\- 8y) are :

„

du
_

Bv .

du
-

(i)

Of the terms on the right-hand sides the first represent translation of
the element as a whole, which can give rise to no internal friction.
The remaining terms express the velocities of Q relative to those of
G. We have to deal only with these relative velocities, and may

• Proc. Roy. Soc., A. v. 92, 1916.

358 AERODYNAMICS [CH.

shift the origin 0 to G and imagine it to move with the centre of the
element.

The last two terms of the expressions include in general rotation of
the element as a whole about an instantaneous axis through G, with

angular velocity |£, where £ the vorticity =^ ~ (Article 39).

ex cy

Introducing for shortness the symbols —

du , dv „ fdv . du\ ....

(n)

equations (i) give —

8v =

(iii)

and we note that the last terms express rotations such as a rigid body
might possess, which again cannot affect internal friction.

It appears, therefore, that the stresses due to viscosity are
associated solely with that part of the motion which is expressed by
the terms in (iii) involving a, bt c. If $ul9 Svl denote components
along Gx, Gy of this ' motion of distortion ' —

rS' • • H

It is now required to find the principal axes of this motion, i.e.

directions at right

angles such that all
lines drawn parallel to
them within the ele-
ment will be subject
only to simple elonga-
tion or contraction.
Let these axes be Gx',
Gy', and let them
make an angle a with
the original axes. If
in these directions a',
V are component rates

of strain while Sw/, 8w/ are components of the motion of distortion,
the above condition for principal axes is expressed as —

Sw/ == a'S*', 8v/ = 6'Sy . . (v)

Formulae of transformation from the old to the new systems of
axes are readily found, with the help of Fig. 149, to be —

FIG. 149.

Ix] VISCOUS FLOW AND SKIN DRAG 359

8*' = 8* . cos a + 8y . sin a,
8/ = 8y . cos a - 8* , sin a ' ^V1'

and —

8w/ = 8«i . cos oc + 8vt . sin a, ..

8v/ = 8^ . cos a - 8*! . sin a. ' (VU)
Substituting for 8«/, 8*' and 8v/, 8/ in (v) from (vi) and (vii)—
8«! cos a + 8vt sin a = a' (8* cos a + 8y sin a),
8vl cos a — 8ut sin a = V (8y cos a — 8* sin a). ^^
Eliminating 8vlf 8^, in turn from these equations gives —
8Wl = 8^ (af cos» a + 6' sin2 a) + 8y (a1 — V) sin a cos a, r
8wt = 8y (a' sin» a + 6' cos» a) + 8^ (af - 6') sin a cos a. (lx^
Comparison of (ix) with (iv) shows that —

a = a' cos8 a + 6' sin1 a,
b = a' sin8 a + J' cos1 a,
c == i («' — *') sin 2a.

Finally, we have, making use of the equation of continuity —
a + b =a' +V =0,]

a = 0' cos 2a, . (275)

c = a' sin 2a. J

20 1. Application to Laminar Flow

As an important example, consider the simple type of steady
motion consisting of flow in layers everywhere parallel to the plane
xOz and in the direction Ox, the velocity u being a function of y
only. Reference to Articles 24 or 194 shows that—

6-0,

a = a1 cos 2oc = 0.

Since a' is not zero, a *= 45° and the principal axes lie along the
diagonals of an originally square element (Fig. 150). Since _

a' = - 6' = c

lines drawn within the element parallel to Gx* elongate at the rate
J(3w/8y), whJe lines drawn parallel to Gy' contract at this rate.
These rates of strain result from the stresses whose effects upon the
element are fully represented by those of principal component
stresses plt — pt, tensile and compressive, acting parallel to the
principal axes (cf. Article 26).

360

AERODYNAMICS

FIG. 150.

Let us follow what happens to the element during a short time 8*.
First ignoring rotation, it is readily calculated that the #-sides of the

element at A become sloped as at B at an angle 4 — . St. But

simultaneously with the motion of distortion, the element has

possessed an angular velocity = — £.-?. By the end of the time

.-.

interval this has rotated it bodily through the angle — $ — St.

Thus the true orientation and shape of the element after 8* is as
shown at C and, diminishing 8* indefinitely, we see that the #-sides of
the element remain parallel to Ox consistently with the type of
motion assumed in the first place.

202. Expressions for the Stresses

The foregoing analysis removes a difficulty that is sometimes felt
with alternative arrangements of the proofs of Articles 24 and 194,
and justifies our definition of viscosity. With the help of Article 26
we can now write down convenient formulae for the stresses. As
before, a positive sign is taken to indicate tension and a negative
sign compression.

It will be found that the definition of fx requires us to write —

pl = 2jjui' - p

from (275), so that—

Using (34), and subsequent formulae of Article 26 we then find—
P*y = \(Pi - P*) sin 2<x = 2{jic,
pxx = (2(za' — p) cos« a — (2[jia' + p) sin1 a

IX] VISCOUS FLOW AND SKIN DRAG

= 2{ju*' cos 2<x — p = 2{jia — p.

The remaining jy-component of stress is similarly dealt with.
Now substitute for a, bt c from Article 200 (ii), obtaining —

361

. (276)

The first of these verifies consistency with the definition of (A.

203. The Equations of Motion

The general equations for steady viscous flow in two dimensions
are now easily constructed.

Fixing attention on a rectangular element of fluid of sides &x, 8y
whose centre is at G, the component velocities u and v of G will
change as it moves. If Du/Dt, Dv/Dt denote component accelerations,
parallel to Ox, Oyt which become apparent when the motion of the
element is followed, then for steady flow (Article 140i)—

Du __ du du
Dt dx l)y'

Dv _ dv dv
Dt ~~ dy Ul)x
Resolving parallel to Ox, the forces due to the normal stresses on

3px

(i)

the two jy-sides give a difference

. dy, while the tractions on

•*

the two #-sides give a difference -Q? 8y . 8x. The sum of these is

the force on the element in the x-direction, and must equal the
product of its mass and acceleration in this direction, i.e. —

or —

Similarly —

A.D. 12*

Dv _ 9p^
l)t ~~~dy

(ii)

362 AERODYNAMICS [CH.

Substituting for pM3t, etc., from (276)—
Du _ 3*w dp

Dv _ a*« a^> /31?; a*w \

= ~ ~~ *

_
p = ~ ^ ~~ + ** 1

Making use of the equation of continuity (61) to reduce the right-
hand sides and substituting from (i) for the left-hand sides, we have
finally —

du du I dp "

= VV w -- 7T'

V . . (277)

dp '

, a*

where v1 = - -- \- 7— and v = -.
v 8^« ^ 8y« p

These equations may be recast in various forms. For example,
eliminating p by cross differentiation and subtracting, and making
use of (61) and (65), combines them into the single equation for
vorticity —

-I +•!-"*• • • • (278)

Or again they yield a single expression for the stream function <|> :

They may also be expressed in terms of cylindrical co-ordinates.
Thus if wt q denote the velocity components in the directions rt 0,
respectively, so that u = w cos 0 — q sin 6, v == w sin 0 + q cos 0,
the equations transform to —

Dm q*_ 13* V/,M(_«'_2 9q\

__+v^v«' rt r, 80;-

. (280)
Dq wq_ 1 ^ / 9 2 3te;\

» + T ~ ~ 7r ye + v V q r« + f ae/

* The equations (277), in generalised three-dimensional form, are fundamental
to the theory of motion of real fluids and were evolved by Navier, Poisson, de
Saint- Venant and Stokes. The simplified demonstration given is taken from an
article by the author in Aircraft Engineering, Jan. 1933. Another proof on similar
lines has been given by Prescott, Phil. Mag., March 1932. It is also of interest to
derive them in terms of molecular motion, as discussed by Jeans, The Dynamical
Theory of Gases.

IX]

VISCOUS FLOW AND SKIN DRAG

363

These alternative forms are all, of course, exactly equivalent and no
simplification exists in one as compared with another.

204. Extension of Skin Friction Formula

Assuming that we can calculate from the viscous equations, and
the boundary condition of absence of slip, the velocities and pressures
adjacent to central parts of the surface of a long body such as a wing,
Aerodynamic force follows from suitable integrations round the
contour as explained in Article 44. For this purpose it is required,
however, to infer the intensity of skin friction at all points from the
velocity gradients, which will be known, while the only formula we
have (Articles 23, 24) relates to strictly laminar flow, which will not
exist.

FIG. 151.

The curve in Fig. 151 represents part of the contour of a cylinder
in relative motion without rotation parallel to Ox, from which the
angle 6 of the normal drawn outwards from the element 8s is meas-
ured. Distance s round the contour is positive in the sense of 6
increasing, and n outward along the normal. The force on the
element is due to the stresses pxx . . . as shown. If TO is the
intensity of skin friction at 8s, acting in the direction — s, on the
upper surface, we have from the figure —

T08s = — cos 6 (pxy . 8s cos 6 + pyy . 8s sin 6)

+ sin 6 (pxx . 8s cos 6 + pyx . 8s sin 6).

Substituting for pxy . . . from (276)—

. (i)

v ;

^ (sin* 6 - cos« 0)

oy/

21
i ^— — — —

p 8y>

364 AERODYNAMICS [CH.

Now adjacent to the boundary —

a na . . a

— = cos 8 - -- sin 0 — >
dx dn ds

a a a

5- = sin 6 — + cos 6 -

oy on os

and on the boundary 3«/3s = dv/ds = 0. Hence (i) becomes —

5 = (cos 6 ~ + sin 6 ^ (sin8 6 - cos8 0)

V dn dn/ v '

/ 3« a*A

-4- 2 ( cos 67; -- sin 0-r- I sin 6 cos 0
\ dn dn/

— cos 0 - -- h sin 0 — . . . . . (ii)

dn dn

Let q = \/(w2 + v*) ^e the resultant velocity. Differentiating —
dq ^udu vdv
dn q dn q dn

But adjacent to the boundary u = q sin 0, v = — g' cos 0. Sub-
stituting in (ii) —

or the skin friction is obtained from the boundary value of the
velocity gradient along the normal.

This formula is equally significant as giving the skin friction from
measurements of resultant velocity, though we shall find later on
that in experiment a different method is often more convenient. It
must be observed that if on the boundary du/ds, dv/ds fail to vanish,
as, for example, in the case of a rotating cylinder, the formula does
not hold.

It is easily verified that —

the boundary value of the vorticity at the position considered.
Complete expressions for the drag and lift, corresponding to those of
Article 44, may be left as an exercise. They come to —

X • • • V '

the suffix indicating that the integrals are to extend completely
round the contour.

IX] VISCOUS FLOW AND SKIN DRAG 365

VISCOUS CIRCULATION AND CURVED FLOW

205. Suppose a long circular cylinder of diameter d, pivoted
axially in an unlimited expanse of still air, to be given a steady
angular velocity to0. The boundary condition of no slip, together
with the action of viscosity, will generate a motion of the fluid round
it. If this becomes steady, the velocity q at any point in the fluid
must evidently be perpendicular to the radius r.

We will first solve the problem as an application of the general
equations of motion. Since w = 0 and q is independent of 6, most
of the terms of (280) vanish and the equations finally reduce to —

f 1 *P -

~r=-?tr ' ' W

d*q I da q

^+-r-^-5-° • • (n)

the first expressing the requirement that centrifugal force per unit
volume must be balanced by the pressure gradient.

Since q is a function of r only, assume as a solution q = Ar*.
Substituting in (ii) —

n(n - 1) Ar"~* + n- rn~l- r" = 0

Hence the general solution is —

where A and B are constants to be determined by the boundary
conditions, which are : q = \u>Qd, when r = \d, q = 0 when r = oo.
Thus A =0 and B = Jco0^a, so that finally —

or the product qr = constant.

It will be seen that the flow is identical with irrotational circula-
tion round a circular cylinder in an inviscid fluid. Denoting by &>
the angular velocity of any concentric cylindrical surface of the
fluid, we have v = c»>#, u = — <oy (taking the origin at the centre of
the cylinder) and since o = q/r = B/r* —

Bx \ i d ( By

Hence the viscous flow is indeed irrotational.

366 AERODYNAMICS [CH.

With the viscous fluid, however, a moment M per unit length
must be applied to the cylinder to maintain the motion. The
traction is constant round the cylinder, and we may choose to
evaluate it on the jy-axis where the tractional stress in the fluid is
pyK. Hence from (276) —

dv du\

= 2fjto)0. . . (Hi)
For the torque —

M = (27cfT . r)f_rf/2 — 7tpia>o^2.

If q0 is the peripheral speed of the cylinder and R is specified by
q0d/v, we find the following convenient formula for the moment
coefficient —

The above result may be more simply obtained from the considera-
tion that the moment just calculated must be the same for all co-
axial cylindrical surfaces in the fluid. If this were not so, some shell
of fluid would gain or lose angular momentum, which would be
contrary to the assumption of steadiness.

Putting v = eo#, u = — ovy, it is found that the tractional stress
round a coaxial surface of radius r is pr . du>/dr. Hence the moment
is (Jt^Ttr* . d<A/dr and, since this is constant —

do* C
dr ^ r*

where C is a constant, and on integrating —

Applying the boundary conditions evaluates C and E, and substitu-
tion gives (283).

206. Rotating Cylinder within Fixed Concentric Cylinder

This case is readily deduced from the preceding article. Let the
outer cylinder be of diameter D. Then the boundary conditions
now determining A and B are the same at the inner radius but q = 0
when r = \D. Hence —

A - £>' _ rfi - £1

IX] VISCOUS FLOW AND SKIN DRAG 367

and substitution of these values in the expression q = Ar + Bjr
gives —

_ co0<ft /£>* — 4y«\
? ~~ IT \ £>»-<?/

The torque on the inner cylinder, which is equal and opposite to
that on the outer one, is —

M =

giving a moment coefficient on the same basis as before

The case of the inner cylinder being fixed while the outer one
rotates is solved similarly.

207. Curved Flow in Experiment

We have already seen, in examining vortices, Chapter VII, that a
circulatory flow very similar to that calculated in Article 205 can
exist in air. When the fluid is contained between two concentric
cylinders of different sizes, revolving at different rates, qr need not be
constant. The stability of this more general case was first examined
by Rayleigh, ignoring viscosity.

Let it be assumed that an element of mass m and volume V,
initially circulating at radius r1 with velocity qlt is displaced by a
disturbance to a greater radius ra without change of its moment of
momentum. For either radius the condition for equilibrium is,
from Article 205 (i)—

If the force on the element due to the pressure gradient, viz. — V-~
acting inwardly, exceeds its centrifugal force at the new radius,

m ( ~ <7i ) . — , the element will be forced back. Thus the motion is

Vt / Y*
stable if —

or —

f if

368

AERODYNAMICS

[CH.

i.e. if the square of the circulation increase outwards. If this
decrease the motion is unstable.

It is found in experiment that an outer cylinder may be revolved
rapidly round a fixed one before eddying occurs in fluid contained
between them. Rotation of the inner cylinder, on the other hand,
the outer one being fixed, produces eddying at a comparatively low
speed, although viscosity advantageously modifies the foregoing
criterion.* Steady flow may be realised in a well-known type of
rational viscometer the construction of which will be evident, if it is
not familiar.

Rayleigh's investigation may also be applied in principle to
explain a striking phenomenon that is observed in front of stagnation

Turbulence

WIND,

Low Velocity

1-0

f i

FIG. 152. — TURBULENCE SURROUNDING THE FRONT STAGNATION POINT OF A

STRUT.

A similar phenomenon is observed with wings. The contours in the enlarged
lower diagram give : velocity amplitude ; - mean velocity.

points, at least when the oncoming stream is not specially smooth.
Fluid approaching a body exerts centrifugal force towards the
surface, maintaining its path against a pressure drop outwards from
the stagnation point (or line). Particles approaching the surface of
the body closely have their energy reduced by viscosity and, if dis-
placed outwards by a disturbance, find themselves with insufficient
centrifugal force to oppose to the pressure gradient, and so are forced

* Taylor (Sir Geoffrey), Phil. Trans. Roy. Soc., A, v. 223, 1923.

IX] VISCOUS FLOW AND SKIN DRAG 369

out farther. Thus the stagnation point becomes the centre of a
region of weak turbulence extending in front of the body. Farther
round the contour of the body the product qr increases outwards, so
that we should expect stability, and, in fact, the turbulence is damped
out there.

Fig. 152 shows the region of instability and, for comparison, that
of time-average velocity reduction in front of a strut. The enlarged
view gives contours of mean amplitude of velocity variation, as
determined by a hot wire connected with a vibration galvanometer,
and also contours of time-average velocity (R = 2-1 x 105).* The
undisturbed flow in the wind tunnel, in which the experiments were
conducted, was known to be rather turbulent.

APPROXIMATIONS TO THE VISCOUS EQUATIONS

208. The general equations obtained in Article 203 are formidable,
and their solution for flow past a given body, though steadiness be
assumed, presents considerable difficulty. To make use of them
drastic simplification is required, and various curtailed forms have
been suggested as appropriate to different circumstances.

If the velocity be very small and the viscosity large, all the terms
on the left-hand side of (279), having to do with the inertia of the
fluid and not its viscosity, may be neglected, reducing the equation
to—

V4^ = 0 (286)

This approximation is due to Stokes, and the range of Reynolds
number to which it may be applied is known after him. Such
motions are minute, however, even from an experimental point of
view.

Another approximate form, taking considerable though incom-
plete account of the inertia terms, was introduced more recently by
Oseen ; it is —

vv4<J> = t/v* (m*x) - - • (287)

where U is the undisturbed velocity. This equation is appropriate
to the Reynolds numbers of anemometry and has been so employed
by Lamb, Bairstow, and others. Bairstow has also suggested its
value for obtaining rough approximations at somewhat higher
Reynolds numbers and, with Misses Cave and Lang, has developed
integral equations for application to symmetrical cylinders.f

* Piercy and Richardson, Phil. Mag., v. 9, 1930. Cf. also the same authors, Phil.
Mag., v. 6, 1928 /circular cylinder), and A.R.C.R. & M., 1224, 1928 (aerofoil),
t Phi*. Trans. Roy. Soc.t A, v. 223, 1923.

370 AERODYNAMICS [CH.

209. Prandtl's Boundary Layer Equations

The approximation of greatest interest in Aerodynamics is that due
to Prandtl, and depends upon the assumption that viscous effects are
confined to a boundary layer, Article 43, a feature that is character-
istic of most Aerodynamic motions. The process of simplification
consists of examining the relative orders of magnitude of the various
terms of (277) when a thin boundary layer exists, and will be ex-
plained for the case of flow along a flat plate.

Ox is taken in the plane of the plate parallel to the undisturbed
flow and Oy perpendicular to the plate, the origin being at the nose.
On account of the thinness assumed for the boundary layer, y and v
are small compared with x and «, which, together with p, are taken
to be of normal order. Since v is small, it follows that in the second
equation of (277) all other terms may be neglected in comparison
with the ^-term. Hence this equation reduces to —

In Article 42 we found, as a matter of experiment, that the
pressure generated just outside the boundary layer is transmitted
through it to the surface of the body without change. The result
(288) follows equally for curvilinear and unsteady flow. Thus
theoretical justification exists for the experimental result.

Turning to the first equation of (277), we find that the first term
of vaw, viz. 9a«/9#a, is negligible in comparison with the second
term : 9*w/3y*. This is the only simplification that can be made in
usual circumstances, and the first equation therefore reduces to —

du du d*u I dp /^™v

"^r + v^r = v^~a--/- • • 289

dx dy 3ya p dx

On account of the smallness of y, d*u/dy* is large and, if the order of
magnitude of y be denoted by z, it will be of order 1/e1. For all the
terms of the equation to be of the same order, v requires to be of
order eV The thickness of the boundary layer in the jy-direction is
then proportional to Vv or> more generally, to

FLAT PLATE SKIN FRICTION WITH STEADY FLOW

210. Application of Oseen's Approximation

The flat or tangential plate of limited chord (c) provides the easiest
problem of direct Aerodynamic interest. Break-away (Article 159)
does not occur, and the flow is found experimentally to remain steady

IX]

VISCOUS FLOW AND SKIN DRAG

371

up to Reynolds numbers (Uc/v) exceeding 5 x 10* even in moder-
ately turbulent tunnels.

The problem has been solved * to Oseen's approximation in a
general way. Although, for reasons stated in Article 208, this

FIG. 153. — STREAMLINES TO OSEEN'S APPROXIMATION FOR A FLAT PLATE

AT R = 4.

(Figs 153-5 are reproduced by permission of the Royal Society.)

FIG. 154. — VORTICITY CONTOURS TO OSEEN'S APPROXIMATION FOR A FLAT PLATE
AT R = 4, SHOWING THE WIDESPREAD DISTRIBUTION CHARACTERISTIC OF LOW
REYNOLDS NUMBERS.

solution must diverge from fact at the larger scales, it is of
interest to notice some of the results, particularly as they describe in
an approximate manner how a boundary layer comes into being as
Reynolds number increases.

* Piercy and Winny, Proc. Roy. Soc., A, v. 140, 1933

372 AERODYNAMICS

The drag coefficient is found to be given by-

4 1-4839

[Cn.

. (290)

~y \t\,j.\.i z\.

Thus the coefficient is large at anemometric scales. Fig. 153 shows
the streamlines and Fig. 154 the vorticity contours for R = 4, and
evidently no boundary layer has begun to form at this small scale.
Fig. 155 shows in contrast the vorticity contours for R = 4 x 10* ;
the linear scale perpendicular to the plate is magnified in the figure

FIG. 155. — VORTICITY CONTOURS TO OSEEN'S APPROXIMATION FOR A FLAT PLATE

AT R = 4 x 10*.

The linear scale perpendicular to the plate is magnified ten times. Comparison
with Fig. 154 illustrates the growth of a boundary layer with increase of Reynolds
number.

ten times, so that the boundary layer that now exists is very thin.
The drag coefficient is now close to its asymptotic value, the second
term of (290) almost vanishing ; this value exceeds that of mean
experiment by 60 per cent.

At large Reynolds numbers the velocity u at any point x, y, is
given theoretically by the formula —

u = ~ I

V7TJ 0

e~**dz

. (291)

whose values may be written down from tables of the probability
integral. Thus the velocity is then a function of y/Vvx/U only.
If we agree to mark the edge of the boundary layer by 1 per cent,
decrease in velocity and denote its thickness on either side of the
plate by A, then at distance x from the nose —

A, = 3-64Vv*/t7
and at the trailing edge of the plate —

A 3-64
c

(292)

(293)

This result is probably within 30 per cent, of the thickness of the
experimental boundary layer, which is rather thicker.

A point of particular interest may be noted at the trailing edge of

IX] VISCOUS FLOW AND SKIN DRAG 373

the plate in Fig. 155. The concentration of vorticity there may
herald the production of an eddy at larger Reynolds numbers.

211. Application of PrandtPs Approximation

The flat plate problem has been solved to Prandtl's approximation
by Blasius.* This solution, as will be anticipated, is essentially
asymptotic, applying only to Reynolds numbers sufficient for a thin
boundary layer to exist, and cannot be used in the anemometer
range. The drag coefficient (after very slight modification by
Topfer) comes to —

CD = 2-656/Vfl .... (294)

and is only some 6 per cent, less than mean experiment through
the range R = 10* — 5 x 105.

On the same basis as (293) the thickness at the trailing edge of the
boundary layer is given by —

It thickens along the plate in the same parabolic way as in (292).

It may be remarked that the convention adopted above for mark-
ing the edge of the boundary layer is arbitrary, and different writers
use different systems. If a greater percentage drop in velocity is
adopted, the factors in (295) and (293) are smaller.

Instead of discussing Blasius's solution, which is somewhat com-
plex, the problem will be solved approximately by a shorter
method f which is of use in some more difficult cases.

212. Method of Successive Approximation

The flow is assumed to be steady, of undisturbed velocity U, and
R ( = C7c/v, c being the chord of the plate) sufficiently large for a thin
boundary layer to exist. The pressure is then nearly constant
throughout the flow, ignoring the edges of the plate, so that PrandtFs
equations reduce to —

du Su 3a«

u— + v— =v— . . . . (296)
dx dy dy* v '

The boundary conditions are u = v = 0 on the plate and u = U,
i) = 0 at oo.

* Zeits. f. Math. u. Phys., 1908.

I Piercy and Preston, Phil. Mag., Ser. 7, v. 21, 1936.

374 AERODYNAMICS [CH.

If ult vl are known as first approximations to u and v, the
equation —

3#a 3w2 d*u.

«i -^- + vl — = v -—

c# dy 3y*

may be regarded as an equation for determining a second approxima-
tion ut. A corresponding second approximation to v, viz. va, can
then be obtained from the equation of continuity and the boundary
conditions. Repeating the process gives a third approximation.

Successive approximations will not in all problems exhibit the
convergence necessary for success, so that application of the method
is tentative. But if they do, a sufficient number of reiterations
secures what degree of accuracy may be desired in the solution of
(296).

213. Transformation of the Equation

We now make the substitutions u = u/U, V = v/U, so that the
boundary conditions become u = U — 0 on the plate, while at oo
u = 1, V = 0.

Also, we transform the equations from the co-ordinates x, y to
£, Y) given by —

For this purpose we note that —

= . =

3% c ' 3y

and, since it will be found that du/dr^ = 0 gives a solution satisfying
the boundary conditions, that with this simplification —

du __ 3« 3i; 3« STQ __ 5 3w

3* = 3l dx + 3 3i = "

3w _ 3w 3| 3w 3Y] _ /'R

3y ""* 35 3^ + 3^ 3^ "~" v 4 •

whence also —

IX] VISCOUS FLOW AND SKIN DRAG 376

We can write down from the preceding article the equation in %
and y for determining the nth approximation. In terms of the non-
dimensional velocities it is —

a« <». = v**,

— ' 8* + -1 8y 3y«

and on transformation to £, v) it becomes —

It is required to substitute for vn^l from the equation of con-
tinuity, which in terms of x and y is —

3* dy

and transforms to —

=0. . . (298)
Hence —

on integration by parts, the constant evidently vanishing by the
boundary condition on the plate. For convenience write —

Wn-i^- . . (299)

Substituting for vn_1 reduces (297) to the simple form —

-~|j? = /„_! (5) ~/ . . . (300)

Integrating once —

or —

Integrating again —

- + C. . (301)

376 AERODYNAMICS [CH.

Since un =» 0 on the plate, where 5=0, evidently C = 0, and
since «H= 1 at oo we have —

if •>-*«>« ,t . . . (302)

An J °

Formulae for the Skin Friction

Denoting by Fn the nth approximation to the total skin friction
over the plate —

In terms of the new co-ordinates this becomes

or from (301) —

rrA/

^n = V-V V

C Jo \ 7£-o

Hence —

(3°4)

214. Evaluation

We have to assume a first approximation to the velocities and we
take those of inviscid flow, so that ul = 1, vl = 0 everywhere.
From (299) we then have/! (£) = — 2£. (302) gives—

Then from (301) and (304)—

S.-^-fV*1^ .... (306)
V^Jo

^CD1 = 4l(nR)1'2 == 2-257 I?-1/2.

Comparing with Article 2 10, it appears that the second approxima-
tion is the asymptotic solution to Oseen's equation. Tables exist

IX] VISCOUS FLOW AND SKIN DRAG 377

for the integral in the expression for ws, but we shall approximate by
expanding e~^ and retaining only the first term, so that —

Then turning to the calculation of the next approximation w8, (299)
gives —

and from (302) again —

* f

A* Jo

a known integral whose value is (3\Ar/2)* F (\), where F (£) is the
' gamma ' function whose value is 0-894, which we denote for short
by*.

Proceeding in this way, we find that we can summarise the results
of successive calculations as follows :

A - 2

A* - '

- 1 /'£_»Y1/:')" /^ 1
~ AVT/ \3fc

The ultimate approximation obtainable by the present analytical
method is found by putting n = oo, which gives, since the sum to oc
of the geometric series = $ —

Writing CD for the ultimate approximation to the coefficient of the
total skin friction, we find, from (304) —

44 2-734

' ' ' ' (3°6)

a result which is only 3 per cent, in error. Such an error is negligible
for most practical purposes, but a little numerical work serves * to

* Reference should be made to the paper cited (p. 373) for a convenient method.

378

AERODYNAMICS [CH.

take into account the neglected
terms of (305), when successive
approximations are evaluated as
given in Fig. 156 and agreement is
reached at the eighth approxima-
tion with the elaborate solution of
Blasius. Fig. 157 shows dotted
the second approximation (305) to
the velocity through the boundary
layer and also the ultimate result.
The ordinate used permits the one
curve (full line) to be given for
all positions along the plate.
Agreement with experiment at high

1-2

N? OP Approximation

BIG. 156.

Reynolds numbers is quite close, although mean results suggest a
slightly steeper curve. At small Reynolds numbers, or close to the
nose in any case, the dotted line will represent fact. At the nose
itself the distribution of velocity is complicated.

EXPERIMENT AND KARMAN'S INTEGRATION
215. Methods of Measurement

The direct way of finding the skin friction at a point on the surface
of a plate or cylinder is, from Article 204, to estimate the velocity
gradient along the normal. Numerical examples worked out from
the foregoing results indicate that the boundary layers of experi-
ment are very thin ; thus at one-tenth of the way along a plate of

IX] VISCOUS FLOW AND SKIN DRAG 379

1-ft. chord in a stream of 100 ft. per sec. three-quarters of the entire
velocity change occurs within a film 0-012 in. thick. It is seen that,
to estimate the boundary value of the gradient directly, measure-
ments require to be made within one or two thousandths of an inch
from the surface.

Two methods exist for such fine work. In one a wire, of about one-
thousandth inch diameter, and heated by an electric current, is held
parallel to the surface and perpendicular to the stream by a rigid
fork. The fork is fitted with micrometer screws enabling the clear-
ance between surface and wire to be adjusted accurately. Velocity
is estimated from the convection of heat from the wire. A serious
difficulty arises from the cooling effect of the experimental surface
when the clearance is small and the velocity low, and a special
technique * is required to determine rather large corrections on this
score.

The alternative method uses the fractional pitot tube introduced
by Stanton. The specialised form is flat and sunk beneath the
surface, so that only a very narrow louvre with a thin lip projects
into the stream. Photographs and examples of use are given in a
paper by Page, Falkner, and Walker.f Again a difficulty arises, in
that, without special calibration, it is impossible to connect the
projection of the lip with effective distance from the surface ; it
cannot be assumed that the pressure observed with a given setting
refers, for example, to a distance from the wall equal to half the
projection.

Calibration may be effected by mounting the pitot tube with the
projection to be used in a long smooth pipe whose velocity profile is
known.

Example. — A pitot tube 0-4 mm. diameter is calibrated in a
straight pipe of 0-24 in. bore delivering 0-33 cu. ft. of air per min.
When the mouth of the tube touches the pipe- wall, the pressure
observed within it is 0-143 in. water head below the static pressure
at a section 18 in. upstream. Show that the velocity indicated
applies to a position midway between the centre of the pitot tube
and its outer lip.

Since the internal radius of the pipe is 0-01 ft., the mean velocity

0'33

w== =17-5 ft. per sec. Assuming 15° C.,

60 X TC X 0-01 X 0-01 ^ 6

R = 17-5 x 0-02/0-000159 = 2200 and the flow is laminar.
So from Article 195, u]u = 2 — 8r*/D2. Midway between the centre

* Piercy and Richardson, A.R.C.R. & M., 1224, 1928.
t loc. cit., p. 211.

380 AERODYNAMICS [CH.

of the pitot tube and its outer lip, i.e. at 0*3 mm. = 0-001 ft. from
the wall, r = 0-009 ft. and u/u comes to 0-38, whence u = 6-65 ft. per
sec. If the pressure in the tube corresponds to this position,
that pressure = £pwa = \ x 0-105 Ib. per sq. ft. = 0-0101 in.
water head above the static pressure across that section of the pipe.

Now from (272) T = 8pw'//? = — PD/4. Thus — P =32pwa/&°
= 0-53 Ib. per cu. ft. Hence at a distance 1-5 ft. upstream from the
pitot tube section the static pressure is increased by 0-795 Ib. per
sq. ft. = 0-153 in. water. The static pressure is determined in this
position, and the pressure in the pitot tube, situated 1-5 ft. down-
stream, is 0-153— 0-0101 = 0-143 in. water head less, as stated.

With every precaution measurement of skin friction remains a
difficult experiment in which to achieve accuracy. The following
theorem has a particular significance as suggesting a method by
which errors can be minimised, although it also has a wider interest.

216. Karman's Theorem

Without making any assumption as to constancy of the pressure,
let us write down Prandtl's equation (289) in the form —

3aw dp du du

p — pw —-f pv — . (i)

9y2 dx dx dy

and integrate with respect to y through the boundary layer for
any fixed position x. First assuming A const., we have —

the last term of (i) being integrated by parts. Considering this
expression, on the right-hand side the last term is the same as the
first since by the equation of continuity dv/dy = — du/dx ; also,
regarding the middle term, u = v = 0 when y = 0 while u = U,
v = VA> say, when y = A. Turning to the left-hand side, the first
term = 0 when y — A and is equal to the skin friction T when y = 0.
Putting in these limiting values, we have —

(ii)

It is readily shown from the equation of continuity that the
velocity across the edge of the boundary layer is evaluated by —

= -^z\ udy- • • (iu')

IX]

Hence finally —

T = I

VISCOUS FLOW AND SKIN DRAG

(Uu - «•) dy - A ^
3*

381

(307)

if also [7 may be regarded as const. (307) is correct for A varying
with xt additional terms arising on this score finally cancelling out.
But when the velocity (Qt say) just outside the boundary layer
itself varies with x, we have in place of the integral in (307) :—

„ 8 f A 3 f A

Pv ^" udy ~~ p -*r \ u*dy. . . (3070)

CXjv OX J o

This important result t

may be established in ^

another way by considering
the conditions for equilib-
rium of a short length $x of
the boundary layer (Fig. 158).
The force acting on this
section in the direction Ox

is — t8x — 7— Sx . A and

must equal the rate of
increase of ^-momentum
within. This rate is (see
figure) —

u=U

£E&

P*5x

i — )»•

u=O

A T&c B x

[FiG. 158.

Hence —

fD rc rc

— p u*dy + p u*dy + pi/ VA

J A J B J D

- A f "

* ~"~^ *~" A ~ ~~~ fH

(308)

which comes to the same thing as (307) on use of (iii), if 8* be small.

Now the integrals in (308) are very suitable for experimental
determination, and are but little affected by errors in u close to the
plate, such as would lead to large deviations in the boundary values
of the velocity gradient. Thus by exploring round a transverse slice
of the boundary layer perpendicular to the plate we can estimate
closely the mean skin friction on the plate in this region.

Similar remarks may be made from an analytical point of view.
(307) is simpler than (289) and allows of plausible assumptions being
safely introduced regarding the velocity profile when it is desired to
calculate approximate results.

Like Prandtl's equation from which it is derived, the foregoing

382 AERODYNAMICS [CH.

method may also be applied to the boundary layers of cylinders
provided the curvature is not great.

217. Examples. — Measurements of velocity (q) across normals
drawn from two points on the upper surface of an aerofoil in its
median plane : A, a short distance behind the nose, and B, 0-049 ft.
measured round the contour farther downstream, give (n being
distance from the surface in thousandths inch and U the undisturbed
velocity) :

n :

U: (A)
(B) .

0-3

0-82

0-83
1-04

0-94
1-22

1-02
1-38

1-06
1-45

1-07
1-46

1-04
1-41

Estimate* the mean coefficient of friction between A and B.

Plotting shows the data to be inadequate for the method of Article
215 and K£rm4n's theorem will therefore be employed, there being
evidently a boundary layer of thickness (A) 0-006 in. = 0-0006 ft.
Write 8(q/U), etc., for increases ot quantities between A and B at
constant n, and Q' for the mean velocity just outside the boundary
layer. (307a) leads to—

* _ J_ fAr2
tU* - 0-049 ).lu

Some assumption must be made regarding velocities very close to
the surface, although what form this takes makes little difference in
the end. For simplicity, assume q oc n from n =0 to 0-001 in.
Then integrating graphically, the first two integrals come to 0-00020
and — 0-00032, approximately, taking Q9 = 1-23 U.

Now p is independent of n through the boundary layer, and,
applying Bernoulli's equation, just outside it —

where p» is the undisturbed pressure, whence —

Hence the value of the last integral is 0-493 X 0-0005 Ib. per ft.
Finally—

T 10-*
— = — — [0-20 — 0-32 -f (0-493 X ^5)] = 0-0026.

* It should be noted that the estimate obtained is approximate only.

IX] VISCOUS FLOW AND SKIN DRAG 383

The second example given below illustrates the calculation of
approximate values for the skin friction and boundary layer thick-
ness along a flat plate from assumed velocity profiles.

Assume that the velocity profile can be represented with sufficient
accuracy by —

When y — A, u/U — 1 = A — B, giving B in terms of A ; and
whenjy = 0, dufdy = 4C7/A so that, in terms of A and A,

T __ v A ....

pt7«-£7'~A ..... W
Since the pressure is constant for a flat plate, (307) reduces to —

.1- = d

Pt/» dx w \u

Substituting from (i) and integrating gives

where —

f(A) = 0-1071 + 0-13574 — 0-07624*. . (iv)

Combining this second expression for the intensity of skin friction
with (ii) gives the following equation for A —

4 v 3
MA x7iA\'u dx'

J\/i) V

Integrating —

A =

Finally, substituting in (ii) and writing Rx for £7#/v,

' — r7o === L* J\ )J * x * • • \™)

The correct value for the intensity of skin friction follows
immediately from (294), which can be written —

Cn = ? [ ^U dx = 2-656 ( ^

since Blasius's solution is for a semi-infinite flat plate, i.e. one
possessing a nose but no tail, and differentiating —

-, /2. • • • (vii)

384 AERODYNAMICS [CH.

It is not possible to find a value for A which will make (vi) agree
with (vii), but the value involving least error is readily ascertain-
able ; it is A = 1J, giving B = | and T/pt/* = 0-324/f;-1'2, an error
of 2\ per cent.

The nature of the approximation is further illustrated by cal-
culating the thickness A of the boundary layer either directly from
(v) or by combining (v) and (vi), which yields —

With A = 1J this gives A/* = 4-64J?71/a. The result can only
roughly be compared with (295), since u/U tends to 1 asymptotically
with the accurate profile. But evidently the boundary layer for
A = 1^ is too thin, and a somewhat larger value secures better
agreement in this respect, though the corresponding skin friction is
in greater error.

The table below gives the results of evaluating by the above
method some suggested alternatives to (i). y± is written for jy/A .

ujU =

(rlPU*).RxW

(A/AT) . J?*l/2

2yi-:vi2 .

2^i - 2^ + yf
sin JTT^J

0-365
0-343
0-328

5-48
6-83
4-79

The foregoing method may easily be formulated in general terms.

218. Transition Reynolds Number

A number of experimental investigations has been carried out on
the skin friction of flat plates in steady flow, but, not unexpectedly,
these fail to agree closely with one another ; the mean observations
of a single experienced investigator may vary by as much as ± 7 per
cent. Some different sets of observations, roughly averaged, are
given as three experimental curves in Fig. 159, and there compared
with the foregoing theoretical solutions. Curve (2) is excessive for
R > 30 ; (1) under-estimates for R < 106. Evidence so far avail-
able points * to the empirical formula —

CD = 2-80 #~1/2 .... (309)

as representing mean experiment under steady conditions at large
scales.

* Page, A.R.C.R. & M.f 1508, 1933.

IX]

VISCOUS FLOW AND SKIN DRAG

385

log R

OK>

FIG. 159. — THE FLAT PLATE WITH STEADY FLOW.

(1) Prandtl (Blasius's solution) ; (2) Oseen (Piercy and Winny's solution). Ex-
perimental : (3) Fage, Miss Marshall, .... Hansen.

If a thin flat plate be held tangentially in a wind tunnel and the
speed increased, the flow within the boundary layer is at first
steady, but at some scale, depending on initial turbulence (large for
a smooth stream) and shape of nose (large for a sharp leading edge)

SL .

DC/C

FIG. 160. — PASSAGE TO TURBULENCE IN THE BOUNDARY LAYER OF A FLAT

PLATE.

the flow within the boundary layer becomes unsteady near the
trailing edge. The Reynolds number, based on the length c of
the plate, at which turbulence just sets in, varies from 10* to
5 x 108 if atmospheric steadiness be included. As speed is further

A.D.— 13

386 AERODYNAMICS [Cfi.

increased in a given case, the position at which streamline flow fails
creeps forward. A large increase in friction occurs there. This
effect is well shown by measurements of Burgers and Zijnen * from
which Fig. 160 has been prepared. Thus at higher Reynolds
numbers a front part of the boundary layer is steady (or laminar in
the accepted sense of the word) and the remaining back part
turbulent. The passage from laminar to turbulent flow in the
boundary layer is called transition and the position at which it
occurs the transition point. If this point is distant x from the nose
of the plate, Uxjv is called the transition Reynolds number. It is
not easy to obtain measurements that are quantitatively consistent
or to explain completely such variations as occur. But it may be
assumed that under constant conditions the transition Reynolds
number would be constant for wide variation of x/c. The same
phenomenon occurs in the curved boundary layer of a thick body
and the same definitions apply, x being measured round the profile.

2i8A. Detection of Transition

The above method of measuring transition Reynolds numbers is
laborious and others are in use as follows.

(1) One method, developed at Cambridge, depends upon the great
increase which transition causes in the thickness of the boundary
layer. A fine pitot tube is located in the turbulent part of the
boundary layer and moved gradually upstream at a constant distance
from the friction surface. If a suitable clearance has been chosen,
the tube emerges from the boundary layer at the transition point
into potential flow, showing a rise of pressure.

(2) Another method, avoiding all disturbance of the flow, has
been developed at Queen Mary College, f and consists of burying a
very small microphone beneath the friction surface, communicating
with the boundary layer through a small hole drilled in the position
where transition is likely to occur. Error in this position is corrected
for by adjusting the tunnel speed or other means. The transition
point fluctuates slightly, causing rapid pressure changes which
become audible on suitably connecting the microphone to an
amplifying set.

(3) The foregoing have recently been superseded by a visual
method devised by Gray J at the R.A.E. for flight tests. In the
form developed at the N.P.L. for use in tunnels, aerofoils are coated
with an emulsion containing china clay and sprayed before a test

* Dissertation, Delft, 1924 ; scales are not given in the figure, since criticisms can
be directed against the numerical accuracy of these early results.

f Winny, Ph.D. thesis, London, 1031. J A.R.C. Report Ae. 2608, 1944.

IX] VISCOUS FLOW AND SKIN DRAG 387

with nitro-benzine, which has much the same refractive index and
makes the white coating temporarily invisible. The nitro-benzine
evaporates more quickly in turbulent than in laminar flow, and thus
the white coloration first reappears under turbulent parts of the
boundary layer. Other expressions of the device are also employed.

FLAT PLATE FRICTION, TURBULENT FLOW
219. Thickness of Turbulent Boundary Layer

Although turbulent boundary layer flow is familiar in Aeronautics,
it is not, unfortunately, amenable to analytical treatment, and
examination depends ultimately upon experiment. Independently
of one another, Prandtl and v. K£rmdn established semi-empirical
laws, known as the power formulae, expressing the application to
plates of experiments in pipes, which are easily carried out with great
accuracy. As before, we choose the origin at the nose of the plate,
Oy perpendicular to the plate and Ox in the direction of the un-
disturbed velocity [7. The velocity u within the boundary layer
will mean the time-average value at any point.

The underlying assumption is that u is expressible in the form —

. (310)

where A is the thickness of the boundary layer and n a constant. On
analogy with Article 197 it is further assumed that, through a
certain range of R, n = 1/7. We adopt this index with the under-
standing that it can be varied afterwards.

Denote, as before, the local skin friction on one side only of the
plate at distance % from the nose by T. At large Reynolds numbers
the pressure p is constant with turbulent as with streamline flow to
a high approximation. Thus in (307) the last term can be dropped
and substitution from (310) gives —

Now on substitution from 274), Article 197 (iii) gives —

T =:<.-0228 Pv1/4w7/4y-1/4

for the pipe friction in turbulent flow at Reynolds numbers such that
the seventh-root velocity formula holds. This is independent of
the radius of the pipe (as originally assumed) and substituting from
(310) reduces it to —

388 AERODYNAMICS [CH.

Equating the two expressions for

A'^- 0*86

Integrating —

/4
J A8/* =0-235

or —

A = 0-375

/VY'B

f — j x'1* = kx\ . . (313)

for constant fluid and speed.

This result should be compared with (295), which may similarly
be written A = k'x*. The turbulent part of the boundary layer
increases in thickness much more rapidly along the plate than the
streamline part.

220. Total Drag Coefficient

We first assume the boundary layer to be turbulent throughout.
To obtain the coefficient of the total skin friction, we double T, in
order to take both sides of the plate into account, and integrate
from nose to trailing edge.

Using (313), (312) becomes—

Now integrating —

l'»xf

0-144

where R = Uc/v.

This drag coefficient is much greater than that for streamline flow
at the same Reynolds number. Taking for example R = 4-9 x 106,
when different conditions would make the boundary layer ' laminar '
or turbulent, ^/R = 700, Rl!* = 13-74, and CD = 0-0038 in the
former case and = 0-0104 in the latter.

In the general case, as we have seen, the front part of the plate
has a streamline boundary layer with a low mean drag, while the
remaining back part is exposed to turbulence giving a high drag.

IX]

VISCOUS FLOW AND SKIN DRAG

389

To apply through this ' transition range ' Prandtl has suggested for
the drag coefficient of the whole plate —

0-148 3400

1/5

(315)

where again R = i7c/v. This formula contains an empirical increase
of the calculated coefficient in (314) from 0-144 to 0-148 to secure
better agreement with experiment. The other coefficient is deter-
mined by the transition Reynolds number, at which (315) must
agree with (294). The value 3400 is appropriate to transition at
5 X 105 ; it becomes 28,000 for 5 X 106.

221. Check from Direct Experiment

Regarding experimental determinations of skin friction in turbu-
lent flow, it may be noted first that v. Kdrman's theorem will apply
when u is the time-average velocity (we have also to include an
integral for the time change of momentum within the slice of
boundary layer, but this evidently vanishes). Measurements will
usually be made with a pitot tube. Since this is a pw1 instrument it
is quite clear that the pressure within the tube will be greater than
that appropriate to the time-average velocity, but examples show
that the increase is small when, as is usual, the fluctuations of
velocity are of the order of ± 5 per cent.

0-016

0012

0008

0-004

(314

FIG. 161. — THE FRAMEWORK OF FLAT PLATE DRAG AT AERODYNAMIC REYNOLDS

NUMBERS.

390 AERODYNAMICS [CH.

The four formulae : (294), (309), (314), and (315) are plotted
through a practical range of R in Fig. 161. The other curves will
be described shortly.

Numerous careful investigations, some of which are listed * below,
have been carried out with which the foregoing results may be
compared. Variations in the conditions of the experiments,
especially in the shape of the nose of the plate and the degree of
turbulence in the oncoming stream, enable comparisons to be made
with the several formulae. These checks are successful up to at
least R = 5 x 106 provided the coefficient in (314) is slightly
increased as described. The dotted curve to the left illustrates the
change of (315) caused by, for example, an unsuitably shaped or
finished nose or a very turbulent tunnel ; it is obtained simply by
adjusting the second coefficient in (315), as described. Beyond
the above range, the formula (314) would require still further
adjustment to accord approximately with experiment for completely
turbulent boundary layers, and others have therefore been suggested
for high Reynolds numbers, viz. —

n? 11 X4. ^ 0-0612

(Falkner)t CD - -^ .... (316A)

Of these, Karm&a's formula is most frequently adopted in elementary
calculations as it is also successful at lower Reynolds numbers if
the flow is turbulent. A second term may be added for the transi-
tional range. The dotted line to the right in the figure is appropriate
to the exceptionally high transition Reynolds numbers obtaining
under favourable conditions with smoothly constructed and finished
wings in flight. This starts at about the extremity of the range for
the simple formula (314), the accurate application of which therefore
tends to be restricted to wind tunnels and crudely designed or
manufactured wings and other aircraft surfaces, especially those
exposed to the turbulent slipstreams of airscrews. It is desirable to
recall that constant pressure is assumed. Much larger transition
Reynolds numbers could be secured in the absence of initial turbu-
lence by means of a decreasing pressure along the plate.

*Blasius, Ziets. /. Math. u. Phys., v. 66, 1908 (laminar and early transitional range,
smooth flow) ; Gebers, Schiffbau, v. 9, 1908 (late transitional range) ; Baker, Coll.
Res. N.P.L., v. 13, 1916 (entire transitional range) ; Wieselsberger, Gdtt. Ergebnisse,
v. I, 1921 (plates covered with fabric, blunt nose, turbulent) ; Kempf, Werft Reederei
Hafen, v. 6, 1925 (high Reynolds numbers).

f Aircraft Engineering, March, 1943.

IX] VISCOUS FLOW AND SKIN DRAG 391

221 A. Displacement and Momentum Thicknesses

In approximate investigations of skin friction, particularly with
turbulent flow, it is often convenient to introduce two thicknesses,
8 and 6, which are measures of particular properties of the velocity
profile through the boundary layer. Need for the step arises in
the first place from the fact that, though the edge of the boundary
layer is readily located in experiment by the method of Article 42,
the corresponding analytical definition is rather uncertain since the
loss of velocity caused by friction vanishes only asymptotically.

Consider a particular position x along the boundary layer. Let
u be the velocity (or its time-average) at a distance y measured
normally from the surface, and let U be the velocity at the same
point with potential flow. One effect of retardation near the
surface is to push out the streamlines of the potential flow by a
distance 8, say, and we have —

E78 - \Udy - JWy,

i.e.-

where the integration is to extend from the surface sufficiently
deeply into the fluid as to make the remainder negligible. 8 is
called the displacement thickness.

For a flat plate with a laminar boundary layer, 8 may be evaluated
from the curve of Fig. 157. In this case it comes to —

8 / v \1/2

where x is the distance from the nose, and it is thus of the order
A/3. 8 is easily estimated closely from a plausible assumption for
the velocity profile ; thus the profile (i) of Article 217 changes the
constant coefficient of (ii) only to 1-74 with A = 1J.

The loss, due to frictional effects, of momentum crossing the
normal at x can be measured in terms of another length 8 by
equating it to pt/80. Then by Article 216 —

y. (iii)

0 is called the momentum thickness.

392 AERODYNAMICS [CH.

The momentum thickness may also be illustrated, as follows,
with reference to laminar flow along the flat plate with constant
pressure. (307) gives —

T dQ

W^'d* • • • • H
and (vii) of Article 217 gives, approximately,

pC/a 3 \Ux/'
Combining with (iv) and integrating —

e 2 / v v/2

- = 2 (~r (v)

x 3\Ux) ' ' ' ()
and eliminating x gives finally —

2> . . . (vi)

The numerical factor, approximated for clearness, is readily made
more accurate if desired.

22iB. Alternative Form of Karman's Equation

In the case of a flat plate with turbulent or laminar boundary
layers along which there exists a pressure gradient, the asymptotic
velocity Q at x will differ from U but may be calculated from the
normal pressures by Bernoulli's equation. The definitions of 8
and 0 given by (i) and (iii) of the preceding article apply in these
changed circumstances provided Q, which will be a function of x,
is written for U.

Kdrman's equation of Article 216 can then be arranged in the
form* —

9U* dx ' Qdx* ' " ' w

where H = 8/0.

Now Nikuradse f and others have explored experimentally the
velocity profiles for turbulent flow through slightly convergent and
divergent channels. Analysis of these results shows that, though
the profiles vary greatly among themselves, there occurs com-
paratively little variation of H from the value 1-4. It will be
noticed that this value for turbulent flow is much smaller, as would
be anticipated from the change in shape of the velocity profile, than

* Prandtl, Aerodynamic Theory, vol. Ill, p. 108.
f V.D.I., Heft 289, 1929.

IX] VISCOUS FLOW AND SKIN DRAG 393

that for laminar flow, which is found to be 2*59 from the preceding
article, assuming constant pressure.

Adoption of an appropriate constant value for H and other
assumptions enable (i) to be employed in an approximate manner to
to obtain estimations of practical utility, as wiU be described later.

We may conclude with an illustrative calculation based on
Falkner's drag coefficient for turbulent flow along a flat plate with
constant pressure, viz. CD = 0-0612//?l/7. We have —

= 0-0131 l/L

whence, on integration —

0 / v \1/7

- = 0-0153 (— )
x \Ux/

and 8 follows immediately. It will be observed that the procedure
of Article 219 is here inverted.

APPLICATION TO CYLINDRICAL SURFACES

222. We could now proceed to calculate from Prandtl's equations
the skin friction with laminar flow round cylinders of aerodynamic-
ally interesting sections. Such calculations stop at breakaway
(Article 159), or at transition should the latter occur before condi-
tions for breakaway are reached. This development of boundary
layer theory must, however, be left to further reading, which may
begin with the references given below *; the literature is compendious
and specialised, and only a few brief remarks will be made in this
book.

In the solution of the boundary layer equations for laminar flow
over curved surfaces, an assumption must be made as to the
distribution of pressure, and three alternatives are available, Skin
friction will be most reliably estimated from pressures that have
been determined experimentally. As already illustrated, these
differ little from those of potential flow in the case of thin streamline
cylinders, which can always be obtained, or approximated to as
closely as is possible in experiment, by the methods of Chapter VI.
Differences become large, however, for bluff shapes (cf. Fig. 72)
owing to the thick wake. It has been suggested! that, as an
alternative to the experimental pressures, those of potential flow

* Howarth, A.R.C.R. and M. No. 1632, 1934 ; Falkner, A.R.C.R. and M. No.
1884, 1937; Piercy, Whitehead and Tyler (being published).

f Piercy, Preston and Whitehead, Phil. Mag., Ser. 7, vol. xxvi, 1938.

A.D.—13*

394

AERODYNAMICS

[CH.

for an artifically modified boundary might be used in such cases,
the modification consisting of an extension of the cylindrical profile
backwards from the points of breakaway in order to represent the
presence of the wake. Fairly close agreement with experiment is
then secured. On the other hand, an already tedious calculation
becomes still more involved.

The full-line curves of Fig. 162 give the distribution of skin
friction along the laminar boundary layers of the flat plate, the

circular cylinder,* and the ellip-
tic cylinder of fineness ratio 3.
Breakaway or separation is in-
dicated by the position at which
the skin friction becomes zero.
The dotted curve refers to the
circular cylinder with the po-
tential flow pressures assumed.
The chain-line curve indicates,
approximately, the theoretical
solution for potential flow pres-
sures appropriate to a boundary
modified, as described, to take
some account of the wake.

The dotted curve represents
the well-known Blasius-Heimenz

solution and is exact (in accordance with the pressure assumption)
through the range where the dotting is close ; farther away from the
nose it becomes increasingly unreliable, and it cannot be used to
determine the point of breakaway. As the fineness ratio of the
cylindrical section increases, the range of an exact solution of the
boundary layer equations becomes increasingly curtailed, until
soon it extends only a short distance from the nose. Over almost
the whole profile of an aerofoil, therefore, only approximate solutions
of the equations can be found. Of these the oldest and best known
is that due to Pohlhausen, but, though still in use, this has been
superseded for some time where accuracy is required. One of the
more modern approximate solutions is due to Falkner.f The
determination of the separation point is of technical importance,
but in its immediate vicinity the boundary layer equations become
in themselves unsuitable. On the other hand, the rapid decrease
of skin friction in front of this point can be estimated fairly reliably,

* From experiment by Fage and Falkner, A.R.C.R. and M. No. 1369, 1930.
t Loc cit., p. 393 ; see also Falkner, A.R.C.R. and M. No. 1896, 1941.

NOSE BREAKAWAY TAIL

FIG. 162. — INTENSITY OF SKIN

FRICTION.

(a) Flat plate; (6) Elliptic cylinder
of fineness ratio 3 ; (c) Circular cylinder,
experimental ; (d) Circular cylinder,
Blasuis-Heimenz solution with potential
flow pressures ; (e) Circular cylinder,
Piercy, Preston and Whitehead solution
with allowance for wake.

IX] VISCOUS FLOW AND SKIN DRAG 395

and extrapolation leaves little doubt as to the approximate location
of breakaway. In illustration of the physical nature of the difficul-
ties confronting calculation in this region, Fig. 162 A reproduces
the results of an experiment* to investigate the fluctuation of
velocity in the neighbourhood of separation. The cylinder was
circular and the numbers attached to the contour lines in the figure

AUDIBLE LIMIT> (

OVER 60

— ._ AUDIBLE MAX

WIND

FIG. 162 A. — TURBULENCE IN FLOW PAST A CIRCULAR CYLINDER.
The numbers are proportional to the amplitude of the velocity fluctuation

are roughly proportional to the velocity amplitude. Exposing a
fine hot wire, connected to an amplifier, in the shaded wedge of
large velocity amplitude made easily audible the passage of vortices
into the Kdrman trail (Article 160).

222 A. As mentioned in Article 22 IB, the rearranged Kdrmdn
equation there given is suitable for wide employment in an approxi-
mate manner. It has been so used by Squire and Young f to
estimate the skin friction of aerofoils with turbulent boundary layers,
assuming that the small pressure gradients along their boundary
layers at small incidences will not affect appreciably the shape of
the velocity profile, so that the relationship between local values of
T, Q and 6 will approximate to that for a flat plate with completely
turbulent flow and a constant pressure. For further remarks on
the assumptions involved in such applications of this equation,
reference may be made to an article by Prandtl.J The method will
be explained in application to laminar flow.

* Piercy and Richardson, loc cit., p. 369.

t A.R.C.R. and M. 1838, 1938.

% Aerodynamic Theory, vol. Ill, p. 156 et seq.

396 AERODYNAMICS [CH.

Substituting for T from (vi) of Article 221 A, the rearranged
Kdrmdn equation becomes for laminar flow —

dQ 6 d 2 v

where H = 2-59 and Q is the velocity appropriate to potential flow
at a distance x measured round the profile from the front stagnation
point.

This equation can be integrated by means of the substitution

v) r= 60 H -' 2, which leads to —

whence —

or * —

4 v 1 . f* / 0\2[i f 8

°2 = » U W^T 2) Jo \u) dX' ' • (il)

H being a known constant and Q/U an ascertainable function of
x, 6 is readily evaluated, whence T/pC/2 follows from (vi) of Article
221 A in accordance with what assumptions are made. Although
applicability is restricted to small pressure gradients, it is never-
theless of interest to employ the method to estimate the distribution
of skin friction round the circular cylinder. The result is shown as
curve (a) in Fig. 162B, the pressures for potential flow being implied
in the simple formula for Q/U taken from Article 108. The curve
(b) represents the most recent solution for these pressures.f The
Blasius-Heimenz solution is reproduced as a dotted curve. The
curve (a) deduced by (ii) from the very dissimilar one for the flat
plate is seen to have the correct form over the front part of the
cylinder. Greater accuracy cannot be expected without elaboration
in view of the large variation of dpjdx in the example chosen.
More elaborate approximate solutions, taking variation of H into
account, duly yield a position of laminar separation, which the
above first approximation fails to predict. This phenomenon
occurs in a region of rising pressure and retarded flow, where

* An equation of this form with different indices is quoted by Holt, Aircraft
Engineering, 1943, as given by Young and Winterbottom in an unpublished paper.
| Piercy, Whitehead and Tyler (in the Press).

IX] VISCOUS FLOW AND SKIN DRAG

397

2Vft

(bl

30° 60° 90° 120°
ANGLE FROM FRONT STAGNATION POINT

FIG. 162s. — APPLICATION OF APPROXIMATE METHOD TO CIRCULAR CYLINDER WITH

POTENTIAL PRESSURES.

(a) First approximation deduced from flat plate ; (b) Correct solution.

approximate methods tend to lose accuracy, particularly as the
conditions for breakaway are approached.

223. Although the local intensity of skin friction is not easy to
measure accurately, no difficulty arises in determining the frictional
drag of a body since it is only required to subtract from the weighed
drag the drag due to the normal pressures. Many investigations
have been carried out by this means to compare the frictional drag
of aerofoils and streamline bodies of revolution with that of the
flat plate. To take into account variation of surface area for a
given chord or length, results are expressed in terms of a coefficient
CF, defined by —

CF = Frictional drag/lpt/'E1,

where E is the ' wetted ' surface. Reynolds number continues to
be based on the length of the body. Primary reference may be
made to the papers cited,* from which Fig. 163 has been prepared.

* Gdtt. Ergebn., Lfg. 3, 1926 ; Jones (Sir Melvill). A.R.C.R. & M., 1 199, 1928 ; Page,
Falkner, and Walker, A.R.C.R. & M., 1241, 1929; N.A.C.A. Tech. Kept., 394, 1931;
Relf and Lavender, A.R.C.R. & M., 597; Jones and Williams, A.R.C.R. & M.,
1804, 1937.

398

AERODYNAMICS

[CH.

0-008

0-006

0-004

0-002

FIG. 163. — EXPERIMENTAL FRICTION IN RELATION TO THE FLAT PLATE FRAMEWORK.

(1) Thin wings ; (2) thick wings and struts ; (3) flat plate by extrapolation ;
(4) and (5) airships ; (6) airship with wholly turbulent boundary layer.

These well-known investigations were carried out before the advent
of low turbulence wind tunnels or laminar flow wings, but remain
worthy of consideration both for their various aspects of permanent
interest and also because (a) most wind tunnels are still fairly
turbulent, (b) average wing construction falls rather short of
theoretical requirements for maximum delay of transition.

Curves 1 (N.P.L. and Gottingen), representing ordinary sym-
metrical aerofoils of 5-6 per cent, thickness ratio, follow fairly
well the transitional drag curve for the flat plate realised experi-
mentally by Gebers. Still thinner aerofoils show a smaller friction,
and the N.P.L. experiments allow of the prediction of flat plate
friction by extrapolation on the assumption that it will be the
same as for a symmetrical aerofoil of zero thickness ; the curve 3
is obtained in this way. The hatched area 2 includes strut sections
of 27-40 per cent, thickness (N.P.L.). It is seen that the frictional
drag of aerofoils is greater than that of the flat plate, but not
greatly so if allowance is made for earlier transition with thick
sections. The extension of laminar flow in the boundary layers of
thick sections is discussed in the next chapter.

Turning to streamline bodies of revolution, the hatched area 4
refers to a model of fineness ratio 5|, and curve 5 to the airship
R 101. These suggest remarkably little change of CF but a greater
tendency under three-dimensional conditions to maintain steady flow.

IX] VISCOUS FLOW AND SKIN DRAG 399

On looping a thin string round the nose of the model giving curves 4,
CF changed to curve 6, the entire boundary layer becoming turbulent.
Curve 6 agrees in an average way with tests on another model
(N.A.C.A.) with turbulent boundary layer. The same change can
be effected for any streamline body by means of a turbulence-
producing screen located upstream, and cannot be avoided with an
airscrew in front of the surface.

Another matter of importance emerges from the many experi-
ments of the above kind that have been carried out in various
laboratories. If a model is suspended by a wire attached in a
laminar flow region, a notable increase of friction occurs, though
insufficient to accord with a wholly turbulent boundary layer. We
conclude that a wedge of turbulence exists behind the wire, while
at laterally displaced positions the flow remains streamline. A
similar effect is caused by sharp longitudinal edges or ridges ; if
the edges are widely spaced, the strips of turbulence will have
limited lateral spread, though the increase of total friction may be
considerable.*

It will be appreciated that through a very wide range of Rt tests
on the same model in different wind tunnels with different degrees
of initial turbulence will disagree. Tests in a given tunnel usefully
compare one model with another, but can be applied to design
only when the effective turbulence is known. Since a curve of
type 6 is easier to extrapolate to full scale than a transitional curve,
whilst the latter depends acutely on initial turbulence, some
designers having access to only small wind tunnels have in the past
deliberately increased their turbulence. But the modern trend is
towards exceptionally smooth streams, with large Reynolds secured,
if necessary, by two-dimensional testing. This matter is returned
to later on.

DEVELOPED TURBULENCE AND ROUGHNESS
224. Reynolds Equations of Mean Motion

The semi-empirical formulae of Articles 197, 219, and 220 have been
noted to be subject to rather rapid change with increase of Reynolds
number, which at much larger scales is less marked. The turbulence
is then said to be more fully developed and may be expected to be a
little easier to analyse. Moreover, this stage is approached with
modern aircraft. The following articles merely introduce what is a
wide and difficult subject whose threshold has scarcely yet been

* These and similar effects can now be demonstrated visually by method (3) of
Article 2 ISA.

400 AERODYNAMICS [CH.

passed by research. We begin with an extract from a notable
pioneering paper by Reynolds.*

Referring to Article 203, we must add for unsteady flow to the
right-hand side of the first of equations (i) the term du/dt and to that
of the second dv/dt. Using the equation of continuity, the first of
equations (ii) can then be written as —

p — = — (pxx — pUu) 4" ^~ (Pyx — Puv) • (317)

and the second similarly. These equations are exact.

Now let u, f be the mean values of ut v at any point and #', v' the
added fluctuations, so that at any instant there —

u = u 4" «', t; = v + v'

It is then necessary that, if the mean fluctuations be reckoned in the
same way and indicated by a bar, u' = 0 = vf. For rapid fluctua-
tions—

uu = u* 4" 2«S' -f- u'u' = w* 4- u'u'.

Similarly,

uv = uv 4- **'*>'•

Substituting in (317) we find equations of mean motion, the first
being —

3w 3 •— 9 — — -

p Ji = ai $" ~ p"f ~ ptt'w/) + 8>i ®» ~Pm~ PM/U') • (318)

Comparing these approximate equations for turbulent flow with
(317), it is seen that additions to the stresses, represented by the last
term in each of the brackets, are caused by the turbulence.

225. Eddy Viscosity and PrandtPs Mixing Length

Although viscous stresses co-exist with the turbulent stresses just
found, it is assumed from experiments with pipes (cf. Article 197)
that through the bulk of the flow the former are comparatively un-
important and may be neglected. This has the disadvantage that
we cannot approach the boundary, where viscosity predominates,
but further development is hardly concerned with establishing a
mathematical theory, which has proved a difficult task, but rather
with inferring from observation approximate laws of sufficient
generality for use beyond the realm of the original experiments.
It is also assumed that the turbulent additions to the normal stresses

* Phil. Trans. Roy. Soc.t 1894 (see Lamb's Hydrodynamics).

IX] VISCOUS FLOW AND SKIN DRAG 401

Px*> Pyy> are of small account in determining the character of the
motion compared with those to the shearing stresses. Thus we
write approximately —

ryx = - p«V (i)

and investigate — in the simplest possible circumstances, choosing
mean motion parallel to Ox, say, with u a function of y only — the
turbulent analogue of two-dimensional laminar flow. The fluctua-
tions are treated as if they were two-dimensional, though actually an
originally two-dimensional steady flow becomes three-dimensional on
developing turbulence.

After Boussinesq, the following formula may be framed on
analogy with the definition of the physically constant viscosity pi :

du .

V=«^- - . . (u)

e is called the eddy, or sometimes the mechanical, viscosity. Calcula-
tion from experimental data shows e to be much greater than jx, as
expected, and not a physical constant.

Prandtl has drawn a parallel between the interchanges from one
layer to another of the molar masses (or particles) in turbulent flow
and those of molecules in laminar flow, substituting a mixing length I
(in the jy-direction) in place of the mean free path of the molecules
(cf. Article 23). It must be observed that in the kinetic theory of
viscosity the molecules clearly suffer no change of momentum while
describing their paths, but that a corresponding immunity cannot be
supposed for particles. This point will be returned to. Meanwhile,
it is assumed that ^-momentum is conserved during the time of
transference and I is regarded as a mean path consistent with this
assumption. Suffix 0 indicates mean absolute values.

Then a particle penetrating a transverse distance / causes a change
of velocity u' at the new position and this is equal to l(du/dy) if / be
small. Hence —

— du

ryx = - pvV = - pi/. / — . . (iii)

Now Prandtl assumes that i/ is induced by opposite values of u1, pro-
portionately great, whence —

the mixing length being adjusted, if necessary, to absorb any co-
efficients arising ; its magnitude is not determined, and does not
remain constant under given physical conditions.

402 AERODYNAMICS [CH.

226. Returning now to the question raised in the preceding
article, G. I. Taylor * has suggested that, while the ^-momentum of
the particles may change during transference, their vorticity will
remain constant, if in fact viscous effects are negligible as assumed.
Exchange between the layers of the mean flow of vorticity rather
than of momentum leads to a different scheme for determining a
mixing length from experimental data. He obtains the equation —

do , y dzu

-,- = - pu'. / -j-i. . . . (320)

dx ° dy* v '

It is not yet easy to decide from experimental evidence completely
in favour of the one scheme or the other.

227. Kdrman's Similarity Theory

In order to carry (319) further v. Kdrmin has introduced the
hypothesis that in every region of the turbulent motion the local
flow patterns are statistically similar, scales of time and length only
varying. Then a first approximation to / is obtained as —

r=7i .... 321)

*u/dy* v '

where x is a number. Substituting in (319)—

, v

' • • <M2>

A more convenient form of this result is (dropping the suffix and
bar)—

r- (duldyY

*i* = *-*$>- • • • <828>

The quantity on the left-hand side has the dimensions of a velocity.
Its boundary value, viz. VWp), is often referred to as the friction
velocity.

It is not yet known how far the similarity assumption can be
justified, but v. K&rm&n f has applied it to the case of turbulent
flow through pipes of circular section. In this case T is proportional
to radius, and replacing it by the skin friction enables (323) to be
integrated, giving approximately —

tt = ^llog| . . . (324)

* Proc. Roy. Sac., A. v. 135. 1932.

f These results are taken for the most part from a paper in the Proc. Internat. Con.
f, App. Mech. (Cambridge), 1934, to which reference should be made ; original
publication was in 1930.

IX] VISCOUS FLOW AND SKIN DRAG 403

where 8 is a constant of dimensions L. With a smooth wall 8
depends on TO, p and v, and may be replaced by the quantity :
const, x V/>V/(TO/P), whence —

/(Wp) /,
-- — (10g

X \

,
- + const.

(326)

This logarithmic formula is suggested in place of the corresponding
power formula already considered when very high Reynolds numbers
are concerned.

Carrying over to flat plates with the help of a further assumption
explained in the next article, v. KclrmAn finds the approximate
formula :

V(2/CF) = - log (7?CF) + const.
x

(326)

Making use of experiment —

1/VCF = 4-15 loglo (RC¥) — 1 . . (327)
where CF is defined in Article 223.

Some other results that can be deduced are in good agreement
with recent experiment at great Reynolds numbers. The value of x
deduced from observation appears

to vary between 0-36 and 0-41 ; the MAXIMUM FOR

value used in (327) is 0-39.

228. Skin Drag

An assumption involved in the
preceding article is that for turbu-
lent flow through pipes the quantity :

um ~— u . t

where um is the maximum

mean velocity (at the axis) and u
the mean velocity at radius r, is a
function of v\a only, a denoting the
radius of the pipe.

Experiments with rough pipes
give greater resistance for a given
flux than with smooth pipes, or,
put another way, a rough pipe exerts
the same resistance TO as a smooth
one of the same diameter when the
mean velocities at the same radii
are much smaller. The above as-

0-5

FIG. 164. — VELOCITY PROFILES (EX-
AGGERATED) FOR SMOOTH AND
ROUGH WALLS OF EQUAL RESIS-
TANCE.

404 AERODYNAMICS [CH.

sumption requires the profile across the section of the mean velocity
to be exactly the same even in two such dissimilar cases. A
number of experiments with pipes and channels of different
roughness, beginning with those of Stanton,* show this to be
approximately true except near the walls ; for equal resistance the
velocity defect um — u is the same through the bulk of the stream
for equal values of r/a, though near the walls it may be greatly
different (Fig. 164).

With perfect smoothness TO is determined by the boundary value
of the velocity gradient through the viscous film lining the walls.
The gradient in the vicinity of a sufficiently rough wall is much
less for the same value of TO, the remaining part of the resistance
being due to form drag arising on the protuberances that project
beyond the viscous film. Yet the mechanism of the transverse
transport of momentum appears to remain approximately the same
through the bulk of the flow and independent of the mechanism
by which resistance is communicated to the surface. We have
used TO to denote this resistance however it arises, but with sufficient
roughness it is no longer a pure skin friction but the sum of frictional
and form drag components. This sum is termed skin drag.

A quite different variation of TO with R is found for rough pipes, as
expected. Blasius's and similar approximate laws for smooth sur-
faces may now be distinguished as smooth-turbulent. A pipe that is
slightly rough, i.e. whose surface is, or may be regarded as, more or
less uniformly covered with very fine grains, has a drag coefficient
which at first follows the smooth-turbulent variation, the grains
lying wholly within the viscous film. But at some considerable
Reynolds number it begins to depart from this law, finally approxi-
mating closely to a velocity-squared law. Increasing the grain size
causes earlier departure and a higher final CF.

Formula (325) is inappropriate for asymptotic conditions when
resistance is almost wholly comprised of form drag, being nearly
independent of v, and is evidently determined by some parameter k
specifying the degree of roughness. Karm&n suggests, therefore,
for geometrically similar roughness the formula —

. . (328)

But what precisely is meant by k is not yet quite clear.
More generally, TO will depend on both v and k ; this state appears

* Proc. Roy. Soc., A, v. 85, 1911.

IX1 VISCOUS FLOW AND SKIN DRAG 405

to be realised with well-separated grains or a waviness of surface.
Although TO may then be much greater than for smooth-turbulent
flow, the velocity-squared law need not be approached within the
range so far explored.

FIG. 165. — QUALITATIVE ILLUSTRATION OF
ROUGHNESS EFFECTS.

229. Application to Aircraft Surfaces

Prandtl and Schlichting * have applied tests on rough pipes to flat
plates. Similar effects have also been found in direct experiment in
compressed-air wind tunnels on aerofoils f and airship models,} and
on aeroplane surf aces § in
flight. Fig. 165 gives a
qualitative || view of the
general results of experi-
ments so far published. A
surface is aerodynamically
smooth, if it exerts a pure
skin friction ; i.e. if its
resistance coefficient follows
a smooth-turbulent law.
With high-speed aircraft
this entails a lacquer finish,

but roughness has little (1) Aerodynamically smooth surface (smooth-

meaning divorced from the turbulent friction law) ; (2) wavy or partly
thickness Of the boundary r°ugh; (3) completely rough ; (4) very rough.

layer, or, more explicitly,

that of the viscous sub-layer. Thus, at low speeds, doped fabric
may be regarded as smooth, but a thin boundary layer makes it
Aerodynamically coarsely rough. Waviness effects may be due,
on high-speed craft, to such sparsely distributed roughness as the
remaining projections of countersunk rivet heads.

Aerodynamic smoothness is not easy to secure at the Reynolds
numbers of high-speed aircraft or compressed-air tunnels, and the
increases of drag at stake are important. Thus a grain size amount-
ing to no more than 0-005 in. increases drag by one-half for R =

* Werft. Reed. Hafen, 15, 1934.

f Relf, James Forrest Lec,t Inst.C.E., 1936; Hocker, N.A.C.A. Tech. N., 467,
1933

J Abbot, N.A.C.A. Tech. R., 394, 1931.

§ Schrenk, V.D.L. Jahrb., 1929.

|| Investigations are not sufficiently advanced to take account of several factors
likely to affect the practical case. For instance, according to the explanation given,
roughness might be expected to have greater significance towards the front than
towards the back of the upper surface of a wing.

406 AERODYNAMICS [CH.

5 x 1C6 — 107, At about these Reynolds numbers with speeds
in the neighbourhood of 200 m.p.h. the ' permissible roughness '
for the drag to be a pure skin friction is so small a grain size as
0-0005 in.

If k is the size of each of the granules which may be regarded as
constituting a completely rough surface, of chord c, kfc is the
' relative roughness ' and Rk = t/Jb/v is the ' roughness Reynolds
number/ According to some experiments the drag ceases to be a
pure skin friction at Rk = 100 to a first approximation. Since
thereafter CF depends little on /?, the following provisional empirical
formula has been suggested as fitting some experiments within the
range of Aerodynamic interest :

/ k \
logio iCF = 0-188 ( Iog10 - 10), . . (329)

\ c /

provided Rk > 100, or, since Rk ~ R . Jc/cf provided k/c > 100/R.
These results are given as indicating in a roughly approximate way
the great importance of the effects concerned in connection with
Aerodynamic calculations and must be regarded as of temporary
value ; essential experiments, on which proper analysis depends, are
still in progress.

230. Application to Model Experiment

A model of a small aeroplane wing, for a craft capable of 200 ft.
per sec., could be tested in a compressed-air tunnel under dynamic-
ally similar conditions. R would be 6-3 x 108, approximately.
According to the above, the permissible roughness at full scale would
be O'OOl in. Assuming this quite practical limit to be realised, the
necessity for geometrical similarity would entail reduction on the
model to about 0-00013 in. The requirement of smoothness of this
high order is faced regarding models for compressed-air tunnels, but
calls, of course, for special care in manufacture.

More frequently tunnel tests on model wings are carried out at
R < 5 x 106 and applied at full scale with R > 107. The question
of geometrical similarity is not so urgent, since dynamical similarity
is not attained, but it may be noted that Aerodynamic smoothness
is now easily ensured on the model, but is beginning to present diffi-
culties on the aircraft. At small lift coefficients the induced drag of
wings tends to vanish, and it is frequently necessary to add con-
siderably to extrapolated tunnel measurements of the drag coefficient
in order to take account of greater effective roughness at full scale.

IX] VISCOUS FLOW AND SKIN DRAG 407

Before small-scale measurements can be extrapolated, it is neces-
sary to know, in addition to full-scale roughness, the effective initial
turbulence of the tunnel stream. Thus, taking two extreme cases,
Fig. 163 shows that at R = 5 x 105 the model boundary layer might
be wholly streamline or almost wholly turbulent ; extrapolation of
the friction would be along the transition curve in the first case and
along the smooth-turbulent curve in the second, until roughness
supervened.

The investigations of the present chapter suggest the following
method of estimating the full-scale CD of a wing at zero CL from small-
scale measurements. Measure CD through a range of R including the
highest value obtainable. Determine the normal pressures through
a similar range of scale. Subtracting the proper integration of these
enables CF to be determined. Inspection of the CF variation with
R, together with the actual values obtained for this coefficient, in
comparison with the various laws described above, will show whether
the boundary layer is in a streamline, transitional, or turbulent state
in the given tunnel. With this information it is possible to construct
a special extrapolation curve on the framework provided .by Fig.
161, assuming Aerodynamic smoothness. But it is advisable to
check this prediction by direct experimental evidence from large-
scale tests at zero incidence. Finally, allowance must be made for
full-scale roughness. It is seen to be a question of circumstances
whether CF is greater or less for the model or the full-scale wing,
but the scale effect estimated for the friction may be applied to
the original measurements of CD,* If the form drag is left un-
changed, the estimate of scale effect may be regarded as conserv-
ative ; evidence regarding the implied decrease may be sought
by again experimenting under two-dimensional conditions with
a slice of the wing of 5-ft. or more chord in the tunnel. The
investigation outlined is, of course, laborious, but when a few
examples have been worked out in a given tunnel, inspection of the
CD — R curve alone will often give sufficient information in subse-
quent cases. Collection of such data constitutes what is meant by
gaining intimate acquaintance with a particular tunnel.

Comparisons of observations in a given tunnel with experiments
in full-scale flight suggest the possibility of a so-called turbulence
factor for that tunnel, the assumption being that increase of the

* The turbulence in the oncoming streams of wind tunnels is finely grained and
especially apt on this score to hasten transition in boundary layers. Natural
turbulence in the atmosphere, being characterised by a much larger scale, permits
of delay. Model experiments are frequently corrected, therefore, in anticipation
of a larger transition Reynolds number being realised in free flight.

408 AERODYNAMICS [CH. IX

tunnel Reynolds number by this factor will give an ' effective '
Reynolds number at which agreement with flight tests can be
expected. It is also hoped to determine the factor by some critical
test, such as that on a sphere described in Article 65. Many
questions arise, of which the most important is whether a single
factor could apply to phenomena of different kinds, and the con-
ception remains tentative.

Chapter IX A
REDUCTION OF PROFILE DRAG

230 A. Normal Profile Drag

Profile drag is defined as the sum of skin friction and form drag.
The term is reserved to aerofoils and wings, though the considerations
of this chapter apply in principle to all streamlined bodies.

The corresponding coefficient CDO is expressed on wing area and
may be estimated for a three-dimensional aerofoil by subtracting a
calculated coefficient of induced drag from the total drag coefficient.
The result is approximately independent of aspect ratio provided
this is not very small. Again, for aerofoil sections of the type
universally employed until recently, it is approximately independent
of the lift coefficient provided this is not large. Such sections are
likely to be superseded in the near future but are meanwhile dis-
tinguished by the term ' normal/ Thus each normal aerofoil can
be regarded without much error as having, through a fairly wide
range of flying incidences, a particular value of CDO appropriate to
the Reynolds number, and determinable directly under two-dimen-
sional conditions.

At low Reynolds numbers the boundary layer of an aerofoil is
entirely laminar, but breakaway occurs at certain positions round
the section, depending upon shape and incidence, and results in a
large form drag. At higher Reynolds numbers transition to turbu-
lence usually takes place in the boundary layer before the positions
of laminar breakaway can be reached. Breakaway may still occur
but at least is delayed,* and is commonly prevented altogether at
small incidences by turbulent mixing. In any case form drag is
reduced. Skin friction, on the other hand, is much increased by
transition.

Fig. 165A, based on the results of experiments at the N.P.L. with
a series of Kirmdn-Trefftz sections, indicates the relative magnitudes
of skin friction and form drag at Reynolds numbers in the neighbour-
hood of 1 million. There was apparently no breakaway but only a

* Experimental details for a sphere are given by Page, A.R.C.R. and M. No. 1766,
1937.

409

410

AERODYNAMICS

[CH.

thickened diffusion of vorticity towards the tail. The curves would
be modified by increase of scale, but not very rapidly, and are fairly
typical of normal aerofoils in wind tunnels except at small scales.

FORM DRAG
FRICTION

0004

CD(FORM)
0-002

5 10 15

THICKNESS (PER CENT CHORD)

FIG. 166A. — FORM DRAG OF JOUKOWSKI SYMMETRICAL AEROFOILS
AT R = 106.

0012

The curves of Fig. 165B for CDO at Reynolds numbers of 1, 10 and
20 million, respectively, are averaged results for various aerofoils,
British and American, tested * by the pitot traverse method at zero

lift incidence in the com-
pressed-air tunnel at the
N.P.L. They agree fairly well
with Fig. 165A and may be
regarded as representative of
good normal aerofoils in tun-
nels of moderate turbulence.
The enclosed points in the

aoio

Ot)06

0004

OtK)2

R-107
2xl07

PIERCY AEROFOIL

LAMINAR FIAT PLATt

figure will be referred to later.
Flight experiments on nor-
mal wings show that turbulence
in the boundary layer sets in
more sharply and at a rather
larger transition Reynolds
0*20 "025 number than in the case of
a model in a wind tunnel. The
reduction of CDOto be expected
on this score has been investi-
gated by Squire and Young.f Considering normal sections at a
small lift coefficient, these authors assumed transition to occur at
various distances up to 0-38 chord behind the nose and estimated by
approximate calculation the consequent variation of profile drag.

* R. Jones and Williams, A.R.C.R. & M. No. 1804, 1937.

t A.R.C.R. & M. No. 1838, 1938 ; see also Page, A.R.C.R. & M No. 1852, 1938.

005 010 015
THICKNESS RATIO
FlG. 165B.

IX A]

REDUCTION OF PROFILE DRAG

411

0012

The front parts of the two curves in Fig. 165c (a) give the coefficient

of (single surface) frictional drag for the upper surface only, according

to these calculations. Curve (1) refers to a thickness ratio of 0-14,

curve (2) to a thickness ratio of 0-25, and the Reynolds number is

10 million. The extension of these curves for transition points at

greater distances than 0-38 chord

behind the nose is referred to

later. At (b) in the figure is shown

the change in the distribution of

skin friction following an extreme

displacement of the transition

point for the upper surface of a

normal aerofoil, from close behind

the nose in a very turbulent

wind tunnel or airscrew slipstream

(dotted line) to 0-4 chord behind

the nose in free flight under

favourable conditions (full line).

A change of about half this

amount can often be anticipated,

and then the curves at (a) suggest

a decrease of about 20 per cent.

in the frictional drag. The authors

also found, with the fixed profiles

investigated, a diminution of form

drag with backward displacement

of transition, increasing the above

improvement.

O O2c <Mc O6c

DISTANCE OF TRANSITION POINT FROM NOSE

0008

0006

OO04

0002

0-2c 04c O6c 03 c
DISTANCE FROM MOSE

FIG. 165c.

2308. The Problem of Reduction

Until recently there appeared little promise of substantially less
values of CDO than those recorded in the preceding article. Never-
theless, minimising profile drag is of such importance that research
into the problem has never been interrupted. In addition to testing
large numbers of different aerofoil sections in the hope of discovering
an abnormally good one, more scientific methods have also been
employed for some years, viz. (a) mathematical studies aimed at
determining the optimum shape for a wing section of given thickness,
(b) improvement of the boundary layer flow by operating directly
upon it, mechanically or otherwise.

These two methods are outlined separately below, but the principle
primarily involved is the same. We have seen in the last article

412 AERODYNAMICS [CH.

that a restricted displacement of the transition point effects a
considerable saving in frictional drag. If the restriction can be
removed without undue increase of form drag, the saving in profile
drag is likely to be large. The curves of Fig. 165c (a) have been
extended to illustrate this. Displacing the transition point from
0-2 to 0-6 chord behind the nose, for instance, may be expected to
halve the skin friction.

Various difficulties arise, however, in applying the principle.
Re-shaping the profile for minimum drag cannot be carried to excess
without introducing other disadvantages. Moreover, the practical
feasibility of the method calls for an exceptionally high standard in
the construction and surface finish of wings. The most promising
form of method (b) incurs pump and duct losses which have to be
minimised, and depends in the end upon the mechanical reliability
of plant installed in the aeroplane. In all probability the two
methods will eventually be used in conjunction with one another.

Compressibility is neglected in the following articles. Method (a)
finds further application, however, in minimising profile drag at high
subsonic speeds. This rather different problem is briefly discussed
at the end of the chapter.

LAMINAR FLOW WINGS

2300. Early Example

In applying method (a) to the present problem the primary aim is
to delay both laminar separation and transition so that the latter
only just anticipates the former and occurs as far back from the nose
as is possible without incurring penalties in matters other than
drag.

Laminar separation can be calculated from first principles
approximately, as already described, though the process is rather
beyond the scope of this book ; and the conformal methods intro-
duced in Chapter VI, or other means, can be employed to shape
wing profiles in such a way as to yield far-back positions of this
breakaway. Transition, on the other hand, is less perfectly under-
stood.

Some factors tending to delay both phenomena are easily seen to
be of the same nature, and an important instance is the maintenance
of a negative pressure gradient. Its effect on laminar separation
has already been mentioned (Article 159), while the following
experiment by Dryden illustrates its effect on transition. A flat

IX A] REDUCTION OF PROFILE DRAG 413

plate tested in a moderately turbulent wind tunnel gave a transition
Reynolds number of 1-8 x 108 in the presence of the small negative
pressure gradient caused by the thickening boundary layers of the
tunnel, and eliminating this pressure gradient decreased the transi-
tion Reynolds number by 40 per cent. A qualitatively similar
effect may be expected in flight, small irregularities of the wing
surface taking the place of the initial turbulence of the wind-tunnel
stream in producing disturbances. The magnitude of the negative
pressure gradient necessary to damp out such disturbances, or
prevent them developing, will depend upon their magnitude and
nature and probably upon whether laminar separation or transition
would otherwise result.

In 1939 a Piercy aerofoil of the simple family described in Article
128, with the maximum thickness of its section located at 0-4 chord
behind the nose (see Fig. 92c), was tested at zero incidence in the
compressed-air tunnel at the N.P.L. and gave, at a Reynolds
number of 1 million, the result shown in Fig. 165B as an encircled
point. Owing to differences arising from the method of testing,
either this value of CDO should be increased slightly or the values for
the three curves in the figure reduced slightly. As the point stands
in the figure, the lowest recorded value of CDO for a normal aerofoil
with which it can be compared is shown as the point enclosed in a
square. The improvement achieved by the Piercy aerofoil is thus
not less than some 35 per cent. The advantage disappeared at
large scales but this is known to have been due to initial turbulence
in the stream.*

The measured value of CDO being less than that for a flat plate
(thickness ratio zero in the figure) at the same Reynolds number in
the same tunnel, the improvement was too great to be accounted for
by even the total elimination of form drag, whilst actually the form
drag could not have been less than about one-third normal. It
immediately became apparent, therefore, that the skin friction was
much less, and the transition Reynolds number much greater, than
for normal aerofoils or the flat plate under the given conditions.
This is explained by the exposure of a greater length of profile to a
sufficiently falling pressure, see Fig. 92c.

Aerofoils of the type thus introduced are called laminar flow
aerofoils. To measure their values of CDO may require an exception-
ally steady stream or flight experiments. Wind tunnels specialised
to the purpose are often called laminar flow tunnels and were intro-
duced in America.

*Relf, Wilbur Wright Lecture, R.Ae.S., 1946.

414 AERODYNAMICS [CH.

2300. The laminar flow wing is not a particular design but a
concept exerting a directive influence on the problem of shaping the
profile for minimum drag under given conditions. The conditions
are specified by practical needs or exigencies, which include :
restriction of pitching moment, preservation of maximum lift
coefficient and of control, structural requirements and constructional
deficiencies. Waviness or roughness of surface must not be very
unfavourable ; laminar flow sections serve no useful purpose for
roughly made wings or in the slipstreams of airscrews ; and the
surface must be kept clean.

The process of design is intimate, and potential flow theory finds
an important application in enabling the effects of shape variations,
however small or large, to be determined with accuracy except
towards the tail. Conformal transformation, which in its highly
developed modern form can encompass even such cases as the flapped
wing, facilitates these calculations whilst protecting the profile from
sudden changes of curvature.

The simple family of aerofoils inverted from the hyperbola has
the advantage of reproducing at once a generally satisfactory section
provided the position of maximum thickness is not set back farther
than 0-42 chord from the nose Adjustment of the position within
this practical limit is rendered possible by the fact that the family
has two shape parameters in place of the single parameter of the
Joukowski family.

But the family is insufficient for the full development of the
laminar flow wing, which may require its maximum thickness to be
located nearer to the tail than the nose, whilst the original Piercy
profiles become sharp at the nose when the maximum thickness is
located midway along the chord.

Additional parameters are necessary to remove this and other
restrictions to shape variation. A generalisation to provide nine
or more parameters, if necessary, has been effected by the author and
Whitehead, yielding an exact method for extending potential flow
calculations to extreme variations of profile shape.

With the profile thus made indefinitely variable, the problem of
finding the optimum shape for given conditions is widened. Modi-
fications of shape for examination by potential flow calculations
may be suggested by experiment, if suitable tunnels are avail-
able, or as a result of collateral boundary layer investigations.
Experiment is in any case necessary in some connections, e.g.
to determine effects upon the maximum lift coefficient and form
drag.

IX A] REDUCTION OF PROFILE DRAG 416

In the following brief introduction,* the above generalisation will
be assumed.

23 oE. Incidence Effect

Casual inspection of pressure diagrams shows an acute dependence
upon lift coefficient and the new aerofoils are not exceptional in this
respect. Yet to be of practical interest laminar flow wings must
maintain a long negative pressure gradient through a sufficient
range of lift coefficient to cover ordinary variations of speed, altitude
and wing-loading.

Fig. 165D applies to ordinary flying lift coefficients and shows at
(a) the approximately constant profile drag coefficient of normal
wings, and at (b) the modification produced by a laminar flow
symmetrical section. The interval of CL through which CDO is
reduced is often called the favourable range of lift coefficient, or simply
the favourable range. In the symmetrical case it is restricted to
small lift coefficients, positive and negative, but can be sufficiently
widened, as shown dotted, for application to fin and rudder design.
Again, the mean value of CL in the favourable range can be displaced
from zero, as shown at (c) in the figure, by the addition of suitable
camber.

* So far as is yet generally known, the development of laminar flow aerofoils
during the war was pursued (i) at the National Physical Laboratory, (ii) in America,
and (iii) in the author's temporary research school at Cambridge, The paper by
Relf (loc. cit, p. 413) may be consulted for a description of (i), where it appears
that Goldstein evolved an approximate method, based on Thin Aerofoil Theory, of
calculating profiles which would reproduce pressure distributions specified before-
hand from experimental or analogous considerations ; for the reverse process, the
potential theory of the arbitrary profile, reference may be made to Theodorsen and
Garrick, N.A.C.A. Report No. 452, 1933. It is understood that (ii) relied to a
considerable extent upon experiment in laminar flow wind tunnels. Preliminary
descriptions of (iii) are contained in the A.R.C. Reports : — Piercy, Whitehead and
Garrard, Ae. 1889, 1941 ; and Piercy and Whitehead, Ae. 1890, 1942 : Ae. 2246,
1943 : Ae. 2266, 1943. The procedure in (iii) was to employ the exact method
mentioned in the text above for the potential flow calculations, so that all the wing
profiles belonged to a single, though very extensive, family ; and suggestions for
shape variation were derived largely from mathematical investigations of the
boundary layer flow.

It is too early to compare the advantages and disadvantages, the achievements
and shortcomings, of these various methods, and only the barest introduction can be
given in this book to a development which is prominent amongst those likely to
improve aviation appreciably.

416

AERODYNAMICS

0-81—

TAIL

NOSE

[CH.

Characteristics of the new aerofoils are often exhibited by means
of diagrams of the velocity ratio q/U, i.e. the velocity at the edge of
the boundary layer expressed in terms of the undisturbed velocity.
This system is adopted in the following figures.

The three curves a, b, c, in
the upper part of Fig. 165E
refer to the upper surfaces of
cambered aerofoils. Curve
a is typical of normal sections
and curve b of laminar flow
sections, both at favourable

q I // /f"-\^ ' ' ^ lift coefficients. The differ-

U A// ^/<^\ ence bet ween these two curves

is characteristic. The posi-
tions of the marking letters
throughout the figure indicate
where transition is to be
expected.

As the lift coefficient is
increased further, the nega-
tive pressure gradient implied
by curve b is progressively
reduced and ultimately suffers a reversal near the nose, as indicated
at c. Laminar flow can survive a small localised reversal of a
strongly negative gradient, the boundary layer re-attaching itself
to the aerofoil surf ace after brief separation, but in such circumstances
as are depicted by curve c laminar flow is impossible over the major
part of the profile. Curve a would also become modified to a
forwardly peaked form, and in practice there would be little to
choose between the two sections at the higher lift coefficient.

Curve d in the lower part of the figure refers to the under-surface
of a laminar flow section and again differs essentially from the
corresponding curve / for a normal section. Curve e applies to the
laminar flow section at an unfavourable lift coefficient. But the
lift coefficient for the unsatisfactory curve e is now less than that for
the satisfactory curve d. For skin friction to be a minimum, the
velocity curves on both surfaces should be of the type b and d.
Thus the favourable range is determined by the interval between
the values of the lift coefficient at which the reversals shown near
c and e first become appreciable, the former as incidence is increased
and the latter as it is decreased. Outside this range the profile drag
reverts rapidly to normal.

FIG. 165E.

IX A]

REDUCTION OF PROFILE DRAG

417

It follows that to be effective in flight the magnitude of the
negative pressure gradient must be sufficient not only to overcome
disturbances caused by slight roughness or inequalities of the wing
surface but also to provide for change of incidence.

23oF. Examples of Shape Effects

To illustrate preliminary steps in the design of a successful laminar
flow profile, we may consider in the first place the typical problem
of improving the symmetrical section distinguished by the thin full-
line in Fig. 165F. As will be seen from the corresponding velocity

FIG. 165F.

diagram, this is already a laminar flow section, but we may suppose
that excellence of construction demands a further reduction of CD0.

The position of maximum velocity could be set considerably
farther back by thickening the section a little in the regions of its
front and rear third-chord points. We assume, however, that on
trying this expedient the favourable pressure gradient, though
extended, becomes too weak. It is therefore decided to locate the
position of maximum thickness considerably farther back, resulting
in the section and velocity curve shown by thick full-lines. The
negative pressure gradient is extended without unduly decreasing its
magnitude.

Is the new section satisfactory ? Three instances will be given of
further modifications worth consideration.

(1) The sharp knee in the velocity curve close behind the nose
probably signifies an unduly small favourable range, a matter which
can be investigated by calculating parts of the velocity curves

418 AERODYNAMICS [CH.

in this region for a few small angles of incidence. Assuming that
the knee develops rapidly, it must be rounded at zero incidence,
and this may be achieved by sharpening the nose a little, as shown
by the dotted lines near the nose. There is a consequent loss of
maximum lift coefficient, however, which must be verified to be
negligible or of acceptable amount.

(2) The tail of the new section is blunt, raising two associated
questions. Will the back part of the boundary layer, though
turbulent, break away and increase form drag ; will the power of
the ailerons decrease ? Assuming that special tests or calculations
return rather unfavourable replies to these questions, the section
must be slightly thinned between the trailing edge and the position
of maximum thickness. The new velocity curve is likely to be
improved upon the whole by this change, as illustrated by the
dotted lines towards the tail.

(3) Assuming that the improvement expected under (2) is realised,
would it be suitable to carry the modification further by making the
back part of the profile strongly concave ? This question may be
considered apart from awkwardness of manufacture ; concave back
parts have often been suggested for laminar flow aerofoils. Proper
investigation other than by experiment is complicated, but there is
easily seen to be a risk of unstable flow arising, as described in Article
207, from the streamlines becoming convex towards layers in which
energy has been dissipated through friction.

Camber and Pitching Moment

The amount of camber has a special significance in the case of
laminar flow aerofoils. The mean lift coefficient for the favourable
range and other common requirements being specified in advance,
there is usually little choice as to the camber to employ. Interest
centres rather in adjusting the shape of the camber-line to reduce
the moment coefficient whilst preserving a sufficient favourable
range.

Geometrically, if a fairly thin section requires appreciable camber
whilst its maximum thickness is located nearer to the tail than the
nose, the crest of the camber-line cannot be advanced as far towards
the nose as considerations of pitching moment would suggest without
introducing a concavity, and part of the stabilisation must be effected
by reflexure of the camber-line towards the tail. The loss of maxi-
mum lift coefficient associated with this reflexure may lead to
designs in which the crest of the camber-line is edged so far forward
as to cause a flat on the front part of the under-surface, as in Fig.

IX A] REDUCTION OF PROFILE DRAG 419

165o. The consequent increase of curvature along the front half
of the upper-surface reduces the magnitude of the negative pressure
gradient along that surface, but the gradient along the under-

FIG. 165G.

surface is increased. Fig. 165E illustrates the effect conservatively,
curve b having a less slope than curve d.

The argument may be put another way. Apart from considera-
tions of pitching moment, both the aerofoil surfaces would be given
much the same pressure gradient, and the lift would be more or less
evenly distributed over a large part of the chord. In order to
minimise the moment, lift is added forward of the quarter-chord
point and subtracted aft of that point, a change that clearly requires
the velocity curves to be adjusted in the manner described.

BOUNDARY LAYER CONTROL

2300. The method (b) of Article 230B, known as boundary layer
control, is concerned with improving the operation of existing
aerofoils without necessarily modifying their shape. Laminar flow
sections are included since their development is otherwise limited
by bluntness of tail ; boundary layer control can prevent consequent
breakaway, restricting form drag and conserving the efficiency of
ailerons.

Breakaway results from ' tired ' air in the boundary layer being
unable to proceed very far against a rising pressure. It would
appear feasible to re-energise such air by means of backwardly
directed jets under pressure, but such jets tend to break up. More
usually, therefore, the de-energised air is removed by sucking it into
ducts within the wing. Long narrow apertures or slits may be
located for the purpose close behind regions of expected breakaway,
and their exhausting action, if sufficiently strong brings air un-
affected by viscosity towards the surface to begin <* new boundary
layer. The action can be repeated farther downstream, if necessary.

The same process serves to prevent transition to turbulence, a
new boundary layer being started when transition becomes imminent.
The strong suction required to remove an entire boundary layer
implies large pumping and duct losses. If local suction is used to
prevent transition, thorough scavenging appears to be necessary m
order that the following boundary layer shall be laminar, as shown

420 AERODYNAMICS [CH.

schematically for the upper surface of the aerofoil in Fig. 165H.
But if most of the wing profile is already under laminar flow and
only breakaway is to be prevented, the condition of the boundary
layer behind the slits may be of small importance, provided the
general flow closes in fairly satisfactorily, and this may be achieved
without dealing with so large a flux (lower surface in the figure).
Slits have also been tried in the neighbourhood of the nose.

FIG. 165H.

Alternatively to the use of isolated slits, it has often been proposed
to maintain an exhausting action upon the whole of the boundary
layer through a porous wing-covering. Prandtl * considers the case
in which the distributed action is sufficiently powerful over regions
of increasing pressure as to keep the boundary layer to a constant
thickness and indicates how design calculations may proceed on this
basis.

Subsequent reading must be relied upon for further information.
But it is as well to realise, particularly before starting a research,
that publications on boundary layer control cover only a very small
fraction of the work that has been carried out on this subject in many
aeronautical laboratories during the past twenty-five years. Re-
markable results are easy to produce ; the difficulty lies in
establishing the economy and reliability of the methods by which
they are obtained.

230!!. Many practical applications of boundary layer control
have been concerned with the delay of stalling. This aspect comes
within the purview of the present chapter since stalling usually
results from a rapid forward movement of breakaway, preventing
which not only maintains lift but also avoids large increase of profile
drag. The result may be achieved by suction methods or, as we have
already seen, by slots.

It has often been proposed to apply slots to small incidences in the
case of large wings. The conception leads to a succession of small
wings forming a kind of cascade, Fig. 165i, and the profile drag is the
sum of the skin frictions and form drags of the component members ;
resolved parallel to the direction of motion. In a modern form each

* Aerodynamic Theory, vol. Ill, p. 117 ; 'In all cases it can be proved that an
arbitrary potential flow can be generated by the use of suitable suction methods/

IX A] REDUCTION OF PROFILE DRAG 421

member would be designed for laminar flow with due regard, in the
first place, to the resultant velocity and pressure fields caused by the
other members. The schematic figure indicates that the wake of
each component aerofoil is carried clear above following aerofoils.
Air of undiminished energy is brought to each aerofoil by a process
reverse to that described in the preceding article, avoiding the need

FIG. I65i.

for pumps and ducts. For small profile drag, the complete wing
should comprise a sequence of long laminar boundary layers inter-
rupted by short lengths of turbulent boundary layers, and a series of
small form drags. The external flow is not entirely irrotational but
contains turbulent layers of vorticity, one of the effects of which may
be to produce fluctuations in the boundary layers, calling for
stronger negative pressure gradients.

230!. Prediction of Lift

The advent of wing sections with blunt tails renews interest in
alternatives to Joukowski's hypothesis as a means of predicting the
circulation. Eventual breakaway may be permitted with some
laminar flow wings, and cannot be prevented if the pumping installa-
tion should fail with wings depending upon some types of boundary
layer control. In such circumstances Joukowski's hypothesis can-
not be applied to a thick section with confidence. The problem is
in general difficult but progress becomes possible if laminar separation
can be assumed, a condition formerly realised only at small Reynolds
numbers but which may now be approached in some full-scale cases.

Betz showed that the flux of vorticity across any normal section
of the boundary layer is proportional to the square of the velocity
at its edge. From this theorem and the reflection that in a steady
state vorticity must be transported into the wake equally from the
two sides of the aerofoil, Howarth * proposed that the circulation K
be so determined as to make the velocities just outside the boundary
layer equal at the two points of breakaway. So far, this criterion
has not proved very successful, owing possibly to the necessary

* Proc, Roy. Soc., A, vol. 149, 1935.

422 AERODYNAMICS [CH.

neglect of production of vorticity behind the points of breakaway
and unequally on the two sides of the aerofoil.

Another method has accordingly been proposed * which is founded
upon considerations affecting the wake only, so that all the vorticity
from the aerofoil is included, and is related to the requirement f
that the circulation round any circuit enclosing part of the wake
only and cutting through it at right-angles must be zero. The
application of this method to an elliptic cylinder of fineness ratio

6-4 will be briefly indicated, leaving the original paper to be con-
sulted for further details.

Fig. 165j shows at (a) the approximate positions of laminar
separation at various incidences a. The wake behind these points
varies in thickness, finally gradually expanding by the diffusion of
vorticity .so that streamlines cross into it. Small variations in the
shape and thickness of the wake are found to affect the problem only
negligibly and, as an approximation, the thickness is assumed to be
approximately equal to the projected distance between the points of

* Piercy, Preston and Whitehead, loc. cit., p. 219.

t Taylor*(Sir Geoffrey) Phil. Trans. Roy. Soc., A, vol. 225, 1&25.

IX A] REDUCTION OF PROFILE DRAG 423

separation, and the edges of the wake to be parallel to the stream-
lines. This involves that, along the wake, the velocities at its edges
are equal at opposite points, leading to an unique value for K. The
method is essentially one of successive approximation since the
positions of laminar separation themselves depend upon K.

At (b) in the figure the full-line reproduces the lift curve so cal-
culated and comparison is made with a wind-tunnel experiment on
the cylinder by Page * at a Reynolds number of 0-17 million. It
will be seen that the stalling angle is predicted with some accuracy.
Again, the difference at smaller incidences is partly due to initial
turbulence in the tunnel and the approximate nature of the calcula-
tions. Whiteheadf has re-examined the method in relation to a
cambered aerofoil of the family described in Article 133A and found
good agreement with experiment until past the stall at a sufficiently
low Reynolds number as to ensure laminar flow in the tunnel used.

HIGH SPEEDS

230}. Compressibility effects at subsonic speeds have already been
discussed. Experimental evidence suggests a progressive increase
of CDO, but also that the increment is small until the critical Mach
number is approached. Minimising the increment is of little
importance compared with that of delaying the occurrence of shock.
The shock wave forms near the position of maximum velocity, before
moving backwards with an accompanying change of the pressure
distribution over the profile.

The principal consideration in profile design for high speeds is to
reduce the maximum velocity ratio, which compressibility itself
increases. If the assumption is made that the shock wave forms
when the maximum velocity attains to the local velocity of sound,
the maximum permissible velocity ratio appropriate to incompressible
flow for shock to be avoided at a given Mach number is easily cal-
culated. The result is rather low ; thus an incompressible flow
velocity ratio of 1-20 involves on this basis a critical Mach number of
only 0-73.

The following maximum velocity ratios are typical of laminar flow
sections having a thickness ratio of 0-15 chord :

Camber (per cent, of chord) . . 0 1 2 3
Maximum velocity ratio .. .. 1-18 1-23 1-28 1-33
These figures can be improved upon with a less backwardly displaced
position of maximum thickness or a less pressure gradient, and

* A.R.C.R. & M. 1097, 1927. f Ph.D. thesis, London, 1939.

(6)

424 AERODYNAMICS [CH. IX A

laminar flow sections restricted in this way are more suitable for high
speeds than are normal sections. But for the above thickness the
maximum velocity ratio even for an ellipse is 1-15, and the absolute
theoretical minimum is about 1-14.* Thus there is urgent need for
a drastic reduction of thickness and camber. The thin wing
sections required entail the use of small aspect ratios.

23oK. An aeroplane fitted with wings of a given section having a
sweep-back 6, Fig, 165K (a), can attain without shock a larger Mach

number V/a than the critical
Mach number Uja without sweep-
back. The effect is explained
qualitatively as follows.

A long straight wing in flight
at velocity V and angle of yaw
<];, Fig.l65K (&), may be compared
with the same wing flying without
yaw at a velocity U and with a
velocity of sideslip v. We have
ty - sin-* (v/7), U = V cos <I*
and, away from the wing-tips,

the pressure distribution will be that appropriate to the velocity U.
In so far as 6 can be identified with fy, we should expect the critical
Mach number of the wings as fitted to the aeroplane to be increased
in the ratio V/U = I/cos 0. However, the correspondence is very
rough.

In applying this method to estimate the rolling moment on a side-
slipping aeroplane equipped with a lateral dihedral, we noted that
the calculated moment might be expected to prove excessive by
about one-third, owing to wing-tip and body effects. Additional
cause of error arises with the swept-back aeroplane since the sub-
stitution of sideslip for yaw becomes progressively untenable as the
centre-line is approached ; near the body the streamlines, though
convergent, cannot be deflected appreciably by sweep-back. When
also the restricted aspect ratio imposed by the use of thin wing-
sections is taken into account, little of the above advantage, perhaps
only one-third, may remain. This can still be made considerable,
however, by the use of exaggerated sweep-back. Thus with
6 = 45°, VjU would be 1-14 on this basis.

* Whitehead, A.R.C. Report Ae. 2073, 1942.

FIG. 165K.

Chapter X
AIRSCREWS AND THE AUTOGYRO

231. The Ideal Propeller

Airscrew principles find many useful applications, but for brevity
we concentrate for the most part on propulsion, recognising a need
for modification of treatment in widely different circumstances.
First, following Rankine and the Froudes, we investigate the
characteristics of an ideal propeller of the kind which, like an air-
screw, produces axial thrust by acting on the air passing through its
disc. By what means this action is effected is for the moment of
no concern ; different ' machines ' that may be employed for the
purpose will have different efficiencies ; and to eliminate such varia-
tion the mechanical process is assumed to be perfect. Thus the
propeller is represented vaguely as an actuator disc of diameter D,
over which a thrust T is distributed. The disc will have a velocity
V relative to the undisturbed air (of pressure p) consistent with the
rate at which work is being done in propelling the craft. We also
assume the entire flow to be steady and irrotational (though these
conditions would not be satisfied with an actual airscrew).

The actuator imparts motion impulsively to the air passing
through its disc, and increases its kinetic energy at a certain rate,

which measures the work done

by the impulse, and which for - - -

efficiency should be a minimum.

We then argue, from Article 173, y

that the final velocity through »•

the stream affected should be the h h; h2

same at all points. The flow takes

the form of a jet (Fig. 166), the _

part behind the disc, called the FlG 160

slipstream, attaining a minimum

section, where the velocity is Vs, a maximum. Consider any small
area SS, of this minimum section. A corresponding element of
thrust ST can be calculated from the rate of change caused by the
propeller of the momentum of the air crossing it, or —
8T = P8S,K,(F, - V).

-H* 425

A.D.-

426 AERODYNAMICS [CH.

The condition for minimum kinetic energy then leads to ST oc SS,
independently of the position of the element. Consequently, for
maximum efficiency the thrust must be uniformly distributed, which
we assume to be the case.

Let F0 be the velocity at the actuator disc, and write S for its
area. Then, on momentum considerations, since pSF0 is the mass
flow per second —

T = pSF0(F5 - V). . . (i)

Let pl be the pressure on the face of the disc, and pz that on its
back. Then an alternative expression for the thrust is —

T^Sfa-pi). . . (ii)

Now Bernoulli's equation applies with one constant outside the
slipstream and with another within it. Also the velocity V0 must
be continuous through the disc, so that energy put into the stream
immediately at the actuator is in the form of pressure. Therefore,
applying Bernoulli's equation outside and inside the slipstream, we
have —

P + IfV* =* pl

F0 being regarded as the equal velocity at two adjacent points on
opposite faces of the disc and suffix s denoting the vena contracta of
the slipstream as before. Here, the streamlines being parallel, the
pressure has again become equal to that in the undisturbed stream.
Remembering this and subtracting —

Substituting in (ii) —

T « iPS(7f» - F'). . . (iii)

Comparing this expression with (i), we obtain the important
result —

F0 == \(V9 + V) ... (330)

i.e., of the total velocity added, one-half appears at the actuator.
It is usual to write —

Fo = 7(1 + a) . . . . (331)

a being called the inflow factor. The addition to velocity at the disc
is aV and that at the vena contracta 2aV.

232. Ideal Efficiency of Propulsion

Useful work is done by the actuator at the rate TV, the drag of the
craft, together with any parallel component of its weight, being equal

X] AIRSCREWS AND THE AUTOGYRO 427

to r for steady motion. But the actual rate of doing work equals
the rate E at which kinetic energy is increased in the fluid, assuming
steady speeds, and this is —

E = ipSFo (Vf - F') = \T (Vs + V)
by (i). Hence the efficiency y) is given by —

_ ^

~V) ~lTa' ' ' (332)

The result shows that, other things being equal, efficiency decreases
as thrust becomes concentrated. To express this conveniently,
define a thrust coefficient Tc by T/pF2D2, when (iii) becomes —

and (332) gives—

1 — Y] 2

-^T=-Tc .... (333)

It is important to bear in mind that the ideal propeller should be not
only uniformly loaded but as large as possible for a given thrust.

Examples. — A slow aeroplane weighing 2 tons has an over-all
lift/drag ratio of 8 at 100 m.p.h. What is the ideal efficiency of its
airscrew, of 10 ft. diameter, at this speed ?

The drag = T = 4480/8 = 560 Ib. and pF2 = 51-2, giving Tc =
0-109, whence a (1 + a) = 0-0695 or a = 0-065. Then (332) gives
7] = 93-9 per cent. This efficiency could not be surpassed undei the
given conditions ; an airscrew for this duty might have an efficiency
of 82 per cent., but the further loss would be due to its own character-
istics.

It may further be noticed that since F0/F, = 1-065/1-13, the jet
contracts to a diameter = 0-97 D ; and again that, since (F,/F)a =
1-277, an aircraft part exposed in the slipstream may have its drag
increased thereby in this ratio.

233. The Airscrew

The familiar airscrew imitates the action of the foregoing hypo-
thetical propeller by means of blades whirled round by an engine.
Considerations of efficiency and weight economy limit their number
to two, three, or at most four. The ratio of the total area of the
blades, counting one side only, to that of the disc of revolution
swept out is known as the solidity of the airscrew, and is usually a
small fraction. The blades support, of course, the whole thrust, in

428 AERODYNAMICS [CH.

producing which outer parts are more effective than inner, owing to
their greater speeds ; over an appreciable area surrouiiding the boss
the thrust will, in fact, be zero. Thus we now have to take into
account variation of the intensity of thrust with radius, called thrust
grading. Considering any concentric annulus of the disc, of small
solidity, the distribution of thrust round it is periodic, being concen-
trated only over the blade dements, and the flow through any part
of it pulsates. For the purpose of calculating the flow, we assume,
however, that the thrusts of the several blade elements included in
the annulus may be regarded as distributed uniformly round it, and
deal with a mean flow at the given radius. This is equivalent to
assuming a large number of very narrow blades.

Let the airscrew make n revolutions per second, or its angular
velocity be fi. Each element at radius r (Fig. 167), traces a helical

path of pitch V/n relative
to the undisturbed air.
But the air at the disc is
subject to inflow, and
also has a slight spin in
pIG 1(i7 the same sense as the

airscrew, defined by a

certain rotational factor 6, such that the angular velocity of the air
just behind is 26fi. Of this one-half is due to vortices from the
blades, so that the element has a forward velocity V(l + a) and
an angular velocity Q (1 — b). The angle <f> which the helix makes
with the plane of revolution is then given by —

The thrust of a blade element is derived from its lift, exerted
perpendicularly to its path, while torque arises partly from lift and
partly from drag, which acts parallel to its path. High lift/drag
ratio makes for efficiency, and blade sections are shaped like those of
wings and set at suitable incidences a to their helical paths. <£
increases rapidly towards the boss, and the whole blade forms a
twisted aerofoil. The angle 6 between the plane of revolution and
the chord of the blade at any radius, viz. —

6 = <f> + a . . . . (336)

is called the blade angle.

The axial advance per revolution when a = 0, i.e. <f> = 6, depends
on r, for so do a and £ in (334). But we define the geometric pitch P

AIRSCREWS AND .THE AUTOGYRO

429

of an airscrew as this advance at a radius of 0-35 Dt where D is its
diameter.

By varying the throttle opening of the central unit of a three-
engined craft, we could clearly vary the effective pitch of the central
airscrew through a wide range, and in fact geometric pitch has no
Aerodynamic significance. Such variation is readily carried out on a
model airscrew in a wind tunnel, and thrust measured for all values
of V/n. It is then found that thrust vanishes for one particular
value of V/n for a given airscrew, no matter what V or n may be.
This unique advance per revolution is called the experimental mean
pitch. It is greater than P, because a will be negative, assuming
cambered blade sections. The airscrew must advance a less distance
per revolution, and the difference is called slip, although sometimes
slip is reckoned from P.

It is convenient to define pitch, etc., in terms of D. The non-
dimensional parameter V/nD is denoted by /.

The thrust of the whole airscrew will be denoted by T, and the
torque required to maintain its rotation by Q. Then the efficiency,
expressing the ratio of the rate of useful work done to power supplied,

is-

. (336)

and takes into account all losses, whether inherent in propulsion or

peculiar to the

airscrew.

234. Modified
Blade Element
Theory

Fig. 168 shows
the circumstances
of a blade element
of chord c and
span 8r at radius
r. Its resultant
velocity W is ex-
pressible in alter-
native ways,
e.g.—

W = V(l + a] cosec <£ = rii (1 — b) sec <f>. . (337)
Let $R be the resultant force it exerts, inclined backwards from the

ttr(l-b)

FIG. 168.

430 AERODYNAMICS [CH.

direction of its lift 8L by the angle y = tan""1 (SD/SL), 8Z> being the
drag. Writing 8T', 8Q' for the thrust and torque of the single
element, from the figure —

XT' = Sfi cos (<£ + y), W = 's# sin (<£ + Y)> • (i)
whence at radius r —

V*T l-b
j QS<?' 1 + a ' tan (<£ + Y)' "

Now Lanchester and Drzewiecki introduced the assumption, since
justified by experiment * for important parts of the blade, that
effects of one element on another at different radius (and having in
general a different lift) can be ignored. This enables us to write for
any element SL = CLipPF2(c8r), tan y = CD/CL, etc., where the co-
efficients are obtained by tests in the wind tunnel on aerofoils of the
same section as that of the blade considered and at the same inci-
dence. All induced velocities are included in a and 6, so that tests
should be made under two-dimensional conditions ; it is in this
respect that original blade element theories have been modified.
Since tests are usually made at aspect ratio 6, conversion to infinite
span is required before application to airscrews, and accurate
formulae should be used for the purpose (cf. Article 179) ; scale and
roughness corrections should be applied to the profile drag coeffi-
cients so obtained in accordance with the concluding articles of
Chapter IX.

With this understanding we proceed as follows. A .B-bladed air-
screw has B elements at any radius and, if 8T, &Q denote their
combined thrust and torque, from equations (i) or Fig. 168 —

ST = B (8L cos <£ — SD sin <£),
$Q = rB (8L sin </> + 8D cos <ji).

Substituting coefficients and diminishing 8r indefinitely —

*j- = B (CL cos <j> - CD sin <pftpW*c . . (339)
dr

d^ = rB (CL sin <f> + CD cos <f>) \$W*c. . (340)

W is given by (337), but it will be observed that knowledge of a and
b is necessary to determine <£. With this knowledge, thrust and
torque grading can be calculated at a series of radii having known
sections set at definite incidences. Fig. 169 illustrates practical
thrust and torque grading curves, showing values increasing with rt

* Lock, A.R.C.R. & M., 953, 1924.

AIRSCREWS AND THE AUTOGYRO

431

01 0-2 O-3

r/D

0-5

FIG. 169. — TYPICAL THRUST, TORQUE AND EFFICIENCY GRADING CURVES
FOR AN AIRSCREW.

i.e. with W, until c is narrowed to form a tip. Such curves can be
integrated graphically from boss to tip to give T and Q. Subject to
two corrections — for tip losses and the drag of the boss, depending
on the type of spinner used — the efficiency of the whole airscrew
then follows from (336). Variation of efficiency along the blade is
also shown.

Before proceeding to the calculation of a and b, some points of
interest in (338) may be noted. Ignoring a and 6, it is found by
differentiation that YJ is a maximum when <f> = 45°, approximately,
y then being negligible in comparison with </>, whence maximum
efficiency along the blade occurs at r/D = J/2n. In fixed-pitch
practice r is small for this condition, e.g. taking V = 250 ft. per
sec., n = 25 r.p.s. and D = 10 ft., r/D = 0-16, and little of the
thrust occurs there. The efficiency of the complete airscrew is
increased by decrease of n, or increase of V, keeping n constant, i.e.
by increase of pitch well towards TC. Practical disadvantages exist

432

AERODYNAMICS

[CH.

in the weight of the airscrew and the gearing interposed between it
and the engine, but these tend to become less important with large
fast craft. These results are modified by a and b (which make
general investigation of the maximum possible efficiency of a
practical airscrew more complicated), but the principle still
applies ; decrease of efficiency with large values of £lr/V is funda-
mental to the existence of y, i.e. of form drag and skin friction. That
the lift/drag ratio of sections should be high is asserted directly by
(338). Provided tip speeds do not approach the velocity of sound,
this is only difficult to secure near the root of the blade, where
thickening is required for strength and stiffness.

235. Simplified Vortex Theory

The lift force on each blade is due to circulation round its sections
whose variation with radius casts a vortex sheet from the trailing
edge. This tends to roll up towards the axis and periphery of the
slipstream, but meanwhile extends downstream as a screw surface
which is at first not quite regular owing to contraction of the slip-
stream. Thus the flow approaching the airscrew ceases to be wholly
irrotational on crossing its disc. It must be possible to calculate the
slipstream from the distribution of vorticity within it and the circula-
tion round the blades, if known, and this calculation would yield a
and b. These quantities are now appropriately called interference
factors. To simplify consideration, a large number of blades with
suitably weakened circulation is again assumed. Irrespective of
this assumption, however, it is clear that the flow far downstream
cannot be affected appreciably by the circulation round the blades,
while interference at the disc is due wholly to the trailing
vorticity.

FIG. 170*

X] AIRSCREWS AND THE AUTOGYRO 433

For clearness, first imagine that rolling up is complete at the
blades, so that circulation is constant along them. The simple
trailing system is then a shell of spiral vortices marking the boundary
of the slipstream together with an axial ' shaft ' of vorticity ; it is
illustrated for two blades only in Fig 170. This system can be
resolved into vortex rings and longitudinal vortex lines, the first
producing axial velocity, and the second rotation in the slipstream.
With non-uniform circulation along the blades, the whole slip-
stream will be a pack of such systems. Radial velocities are dealt
with later on.

Now for a lightly loaded airscrew disc we may neglect the contrac-
tion of the slipstream. Then regarding the axial disturbance, it is
clear, from the theory of Chapter VII, that the velocity at any point
in the disc will amount to one-half of that at the same radius far
downstream. This agrees with and extends the result of Article
231.

Considering next the rotation, let us first follow a wide loop A
(Fig. 170), formed always of the same particles of air. Upstream of
the airscrew, its circulation is zero to accord with the irrotational
flow, and this remains true, by Article 14QK, as it threads over the
slipstream, showing that rotation is confined to the slipstream.
Now the circulations of the whirling blades are obviously trying to
cause circulation in front of the disc and, for this to be zero, the
tendency must be exactly balanced by an equal and opposite induc-
tion there by the trailing system. Just behind the disc, however,
the circulations round the blades induce rotation about the axis in
the opposite sense, and this is therefore added to the equal rotation
induced by the trailing system. Thus, if we follow a narrow loop B
(Fig. 170), it is set spinning at the disc, and exactly half this spin is
due to the trailing system. Again, by Article 140K, the circulation
round this loop of particles cannot change as it passes along the slip-
stream, so that, as the influence of the blades decreases, so must that
of the trailing vortices increase. Hence the spin caused in the loop
when at the disc, by the trailing vortices only, amounts to one-half
of that induced by them far downstream. Finally, we remember
from aerofoil theory that modification from two-dimensional condi-
tions is due solely to the trailing vorticity, and deduce that the
rotation interfering with the blade incidence is one-half of that at the
vena contracta.

Summing up, if the velocities of the blades relative to the air they
engage are V(l + a) and &(1 — V then the velocities finally added
are 2aV and 2&Q.

434 AERODYNAMICS [CH.

236. Approximate Momentum Equations

The following treatment proceeds on the assumptions that the
slipstream is sensibly parallel, that rotation is insufficient (as is
known) to cause appreciable variation of pressure, and that averag-
ing round annuli is permissible. For ordinary airscrews, at least,
these assumptions appear to involve little error.

The rate at which fluid mass crosses the airscrew disc at radius r
is m = p . 2nr8r . V(l + a). The velocity finally added in reaction
to thrust is 2aV. Equating thrust to rate of change of momentum —

a)a. . . (i)

Again, the angular velocity finally given to m is 26fi, so
that, considering the rate of change of angular momentum, 8Q
= m . 26fl . r2, whence —

a)b. . . .(ii)

Expressions for the Interference Factors

We must be able to equate the foregoing to the alternative formulae
of Article 234, whence a and b can be evaluated in a given case.

It is convenient to introduce symbols for the resolved force
coefficients and the solidity of the annulus at radius r :

2t = CL cos <£ — CD sin <f>\

2q = CLsin<£ + CDcos<£V . . . (341)

a == Bc/27ir. j

Then equating (i) to (339) gives —

4rcrF2(l + a)a = BtcW*.
Substituting for W from (337)—

— ^— = fat cosec2 f . (342)

In similarly equating (ii) to (340) note that PP2 can be written —

W2 = V (I + a) cosec <f> . Or (1 — b) sec <£.
Then it is at once found that —

-- - = Gq cosec 2<b. . . (343)

1 — 0

These formulae are somewhat awkward to use ; the interference
factors appear in /, q and <£, and graphical or trial and error methods

X] AIRSCREWS AND THE AUTOGYRO 435

are needed to determine them. A suitable method is described
immediately after the next article.

237. Practical Formulae

Defining thrust and torque coefficients for the complete airscrew
in the usual form :

thrust grading is expressed non-dimensionally as —

dkT __dT I _dT 47ra
d(r/D) = ~dr ' pnf& ~" ~dr ' plSD8 '

and torque grading similarly. Substitute for dT/dr, dQ/dr from
(339) and (340) after using (337) to express W in terms of Q. Then —

- • (345)

die / r\4

and- jl =87*1 (^J? (1-6)' sec* #. . (346)

Alternative formulae are obtainable in terms of a.

J for the airscrew is connected with <f> and the interference factors
for the element by the relationship :

from Fig. 168.

The efficiency of the element is given by (338), but may be ex-
pressed in terms of / as follows, making use of the last formula —

1 =•£.-. cot (# + Y). • • • (348)

TC £r

Simpler expressions for t and q than those so far given are suffi-
ciently accurate for most purposes. Since tan y = CD/CL the first
may be written —

2t cos y = CL (cos <f> cos y — sin <£ sin y).
Hence for small values of y —

2t = CL cos (^ + y).

Similarly— . . . (349)

2q - CL sin (<f> + y).

These approximations fail when a for the blade section is near the

436 AERODYNAMICS [CH.

incidence for either zero lift or the stall, and for root sections which
are too thick for high lift/drag ratios.

238. Method and Example

It is usually required in practice to analyse an airscrew for several
flight conditions, i.e. through a range of J. The following procedure
collects the necessary data while meeting the inconvenience of (342)
and (343).

Having selected a radius, subtract from the blade angle 0 a series
of values of a for the blade section, giving a number of <£'s. The
force coefficients will be known for each incidence and tt q follow.
Then a> b, and other quantities are at once calculated and the results
give curves of variation against /. The process is repeated with a
number of other radii. Finally, variation against r/D is read from
the curves at the constant values of J that are of interest and
integration is effected by a planimeter or other means.

A fair degree of accuracy and provision for checking is required,
and it will be found convenient to work out tables of 25 or 30
columns for each radius. Representative columns and rows are
illustrated in Table VIII (an additional significant figure is usually
attempted).

The particular case relates to two-thirds radius of a 2-bladed air-
screw of 9 ft. diameter used on an aeroplane whose top speed is 160
m.p.h., the airscrew then turning at 1200 r.p.m. The pitch/diam.
ratio is 1-5, so that tan 0 = D(P/D) ~ (2n . JZ)) = 1-5 -f- |TU, or
0 = 35-6°. The chord at r = 3 ft. is 0-754 ft., so that a = 0-08.
Zero lift for the section occurs at — 2°, while its lift coefficient slope
is 0-10 per degree. Up to CL = 1*2, y = 1° subject to a minimum
CD =0*010. These aerofoil figures have already been duly corrected
to infinite span. The third row of the table is added for comparison,
taking 10° less blade angle, nothing else being changed.

TABLE VIII

dld(
kt

*ID}:
A«

35-6°
35-6°

1-6°
10-6°

34°
25°

0-146
0-566

0-106
0-276

0-019
0-145

0-009
0-028

1-37
0-83

0-92
0-81

0-153
0-478

0-0367
0-0777

25-6°

1-6°

24°

0-162

0-078

0-041

0-008

0-89

0-140

0-0225

It has been chosen here to illustrate the efficiency of the element.

AIRSCREWS AND THE AUTOGYRO

437

The larger value of 8 gives curve (A) of Fig. 171. Zero thrust occurs
at / = 1-6, approximately, so that efficiency falls steeply. Curve

O03

0-4

1-6

FIG. 171. — TYPICAL EXPERIMENTAL CURVES FOR AIRSCREWS.
The numbers in brackets give pitch/ diameter ratios.

(B) relates to 0 = 25'6° and shows the loss in maximum efficiency
due to the correspondingly smaller value of /. Increase of efficiency

438 AERODYNAMICS [CH.

with pitch is further illustrated by the group of experimental curves
obtained with a family of airscrews. The dotted increase to curves
(A) and (B) indicates the greater efficiency that would follow a
reduction of CD by 20 per cent, to 0-008, which by Fig. 165s is
feasible for normal sections with a smooth, non-wavy surface, at
the Reynolds number for the flight condition (R = 2-1 x 10e) ;
three times this reduction is achieved by using laminar flow sections.
Referring to curve (A), the element practically attains maximum
efficiency at top speed flight (/ = 1-3), but with J = 0-8, which
might correspond to maximum climb, efficiency is much reduced ;
the lift coefficient for the section is then about 1-28 and it is approach-
ing stall, bearing in mind the Reynolds number. That arrest of
thrust and fall in efficiency are imminent are confirmed by the
experimental curves for a complete airscrew of the same P/D ratio
(1-5). The table also shows that a is more important than 6, which
may sometimes be neglected.

239. Principle of Variable Pitch (V.P.)

In the preceding article it has been seen that the representative
blade element (and we may infer the whole airscrew) designed for top
speed is in a poor position to produce climb. In a V.P. airscrew the
blades are socketed into the boss, and 0 is increased or decreased
uniformly along all blades, thus adjusting pitch, either automatically
or under control. Decreasing 6 by 10° increases the local efficiency
in the above example from 0-80 to 0-877 at J = 0-8. However, the
last column of Table VIII shows that the torque coefficient would not
be sufficient to absorb the full power of the engine at its maximum
permissible revolutions, just as with the original pitch it would be too
great to allow the engine to attain its revolutions and develop full
power. When the requisite adjustment is made, there still remains
an important increase in the power available for climb, though with
a light aeroplane of small power it usually requires to be verified
that the rate of climb is appreciably improved after taking into
account the additional weight of the apparatus. But with high-
duty commercial craft, and of course military aeroplanes, which are
usually fitted with supercharged engines for flying at considerable
altitudes, variable pitch is critically important. Apart from
substantially improving climb and maintenance of altitude with
partial engine failure, it may approximately double thrust during
the early stages of take-off, and enable full power to be used with
increase of efficiency at heights where decrease of density would

X] AIRSCREWS AND THE AUTOGYRO 439

otherwise require the engines to be throttled to prevent racing.
As an example, it may be mentioned that the convenient and
efficient twin-engined aircraft design can be carried to considerably
greater all-up weights with V.P. than with fixed-pitch airscrews.

We shall illustrate the use of variable pitch by reference to the
experimental curves of Fig. 171. These do not quite accord with
constant variations of blade angle, but the difference is ignored. An
airscrew of 11 ft. diameter with a two-position hub is assumed,
giving P/D = 1-0 or 1-5. For simplicity, the supercharged engine
of 800 b.h.p. is assumed to maintain its power at constant revolu-
tions (25 r.p.s.) up to 11,000 ft. altitude. The most important
considerations are taken to be speed at this altitude and climb at
ground level, for which first estimates of A.S.I, are 300 and 200 ft.
per sec. respectively.

From the engine data we find that the constant torque exerted by it
is 2800 Ib.-ft. and, taking account of the variation of p (the relative
density = 0-71, approximately, at 11,000 ft.), this gives possible
torque coefficients for the airscrew of 0-0117 at G.L. climb and 0-0166
at altitude. The two values of J are 0-73 and 1-30. It is verified
from the torque curves of Fig. 171 that the two pitches comply with
these conditions. The efficiency at high altitude is 86 per cent, and
during climb at the smaller pitch it is 80 per cent. At J = 0-73 the
efficiency with the larger pitch is only 67 per cent., but the torque
coefficient required would be more than the engine can manage.
640 thrust h.p. is available during low-level climbing and 688 at
high speed at 11,000 ft. where, however, the estimated velocity is
356 ft. per sec. Again taking account of the density, 580 thrust
h.p. would produce the same level A.S.I, at low altitude ; the larger
pitch would be used and the engine would slow down a little and give
less power, but probably more than sufficient to maintain the A.S.L
Provision of a third pitch would improve maximum speed at low
altitude.

Static Thrust

To illustrate numerically the advantage of variable pitch at the
beginning of take-off requires knowledge of the variation of engine
power with rotational speed — information supplied by the makers
from bench tests for each engine. We shall assume the possible varia-
tion : b.h.p. =71-5 w3/4, which accords with 800 b.h.p. at 25 r.p.s., so
that Q = 550 x b.h.p./27m = 6260/w1/4 Ib.-ft. Now at ground level,
still with an 11-ft. airscrew, pZ)4 = 34-8 and pD8 = 383. Assuming

440 AERODYNAMICS [CH.

first the larger pitch (P/D = 1-5) we have kQ = 0-0269 at J = 0
from Fig. 171, whence —

= 0-0269 =

giving n = 17-3. Then since A^ = 0-133 from the figure, T =
1388 Ib. Repeating the calculation for P/D = 1-0 (fcQ =0-0151,
fcT = 0-120 at J = 0), we find n = 22-3 and T = 2077 Ib. Thus the
finer pitch gives nearly 50 per cent, greater static thrust in this
instance. Most of the improvement is inherent in the smaller pitch,
but part is due to the blades not being so badly stalled. This may
be shown in an approximate manner as follows :

When a blade is completely stalled the Aerodynamic force acts
approximately at right-angles to the chord, or y = a. Thus in
equations (i) of Article 234, <f> -f y = 8 and, since P = 2-nr tan 6, or
cos 0 = 2nr sin 0/P, we have for the representative element —

8r'= SR . 2nr sin 0/P = 80' . 27C/P.

If we assume constant pitch we can sum for all elements, obtaining —

T = 27U0/P .... (350)

for a completely stalled airscrew.
Application to the foregoing example gives approximately —

P(ft.) 16-6 11

Q (Ib.-ft.) 3070 2880

T(lb.) 1169 1644

showing an improvement of 41 per cent, in the second case. Neither
of the original airscrews is completely stalled, but the first is the
nearer to that state.

240. Ordinary Tip Losses and Solidity

No mention has yet been made of radial components of velocity
in the slipstream. Though small in its interior, these attain con-
siderable values towards its boundary owing to the small number of
blades, whose circulation diminishes towards the tips. Looked at
slightly differently, the momentum equations require a factor,
depending on radius, expressing that the annulus does not take up
momentum quite so efficiently as supposed. The effect decreases as
the number of blades increases, and a 4-bladed airscrew has an
appreciable advantage in this respect over one of two blades.

Theoretical investigation must be left to further reading ; approxi-
mate treatment has been given by Prandtl * and a closer solution by

* See Glauert in Aerodynamic Theory, v. IV, p. 261, 1935,

X] AIRSCREWS AND THE AUTOGYRO 441

Goldstein.* The simplest way of viewing the results is to conceive
that the diameter of the airscrew is effectively diminished by tip
losses, so that by Article 231 efficiency is decreased at given thrust
coefficients. The fallowing approximate formula results from the
theory to determine Det the effective diameter —

. (351)

This gives the curves in Fig. 172 for two, three, or four blades,
showing that the effect becomes important for large P/D ratios ; at

1-O1

0-8

0-7

O-4

0-8

1-2

1-6

FIG. 172. — EFFECT OF TIP LOSSES.
The numbers refer to 2-, 3- or 4-bladed airscrews.

J = 1-3 the loss in ideal efficiency of a 2-bladed airscrew on this
score is 4 per cent.

Thus tip losses make at least three blades desirable with high
pitch. It may then be necessary, however, to increase solidity,
which through greater skin friction and form drag again decreases
efficiency. This question is easily investigated by the methods
already established, and it will be found that, although efficiency is
much reduced at low P/D ratios, the correction diminishes at
large ratios. It may well result in a given high-speed example that
loss from greater solidity following change from two to three blades
is less than one-half the gain in respect of effective diameter.

* Proc. Roy. Soc., A, v. 123, 1929.

442 AERODYNAMICS [CH.

241. Compressibility and High Speed Tip Losses

Losses of a different kind are caused on the outer parts of blades
by the compressibility of air when tip speeds approach the velocity
of sound. In the example of flight at 11,000 ft. given in Article 239,
the resultant velocity W at r = 0-4D is 777 ft. per sec. neglecting
inflow ; the ratio of this to the velocity of sound at the altitude is
0-72 ; and, without specially suitable blade sections from this radius
to the tip, efficiency forecast from experiments at lower tip speeds
would not be realised. At the compressibility stall, it is difficult
to estimate force coefficients and equally doubtful how they may
be applied to airscrew design as radial flow occurs. Study is more
usefully concerned with avoiding such effects when tip speeds must
be high, than with their computation. The theory of compressible
flow introduced in earlier chapters is supplemented by tests on
aerofoils in high-speed tunnels,* tests on model airscrews rotated

at fast rates in standard
wind tunnels, f on air-
screws in special tunnels , J
and in flight.§ The
following description is
based on these published
data.

The symbol M will
denote the ratio of the
speed of a blade section
at large radius to that of
sound. As M increases,
Glauert's theory suggests
that for moderately high
speeds CL should increase
in the ratio I/ <\/(l — Af1),
and this variation is
realised by some aerofoils .
Drag also gradually rises,
and there is an increase
of thrust and torque, but

with little change of efficiency. When M = 0-7 to 0-8 the
compressibility stall sets in, as illustrated in Fig. 173, several

* See Chapters III and VI B, C.

f Douglas and Perring, A.R.C.R. & M,, 1086, 1927; Hartshorn and Douglas,

A.R.C.R. & M., 1438, 1931.

J Weick, N.A.C.A.T.R., 302, 1928; Wood, N.A.C.A.T.R., 375, 1931.
§ Jennings, A.R.C.R. & M., 1173, 1928 ; E. T. Jones, A.R.C.R. & M., 1256, 1929.

06 07

M
FIG. 173, — THE COMPRESSIBILITY STALL.

(0) Incompressible flow lift coefficient ; (1) 10
per cent, thickness and 10 per cent, camber ;
(2) 10 per cent, thick symmetrical Joukowski
aerofoil section ; (3) Glauert's formula : C\ «

AIRSCREWS AND THE AUTOGYRO

443

reasonable sections falling within the hatched areas. At M = 0*8,
kf may be falling and Y) still more quickly. With greater speeds
these effects are accelerated, and before the velocity of sound is
reached by the tips the following results are not impossible in a
bad case : CL down well below the incompressible flow value,
perhaps by one-third ; CD approaching 0-2 ; y passing through
20° ; thrust diminished by 10 per cent., and efficiency by more
than 20 per cent.

In spite of these alarming figures airscrews can be designed to
attain M = 0-9 at the maximum radius without loss of thrust and
possibly also with little loss in efficiency, see Fig. 173A. The first
precaution is to restrict thickness ratios
for outer sections ; 8 per cent, is a
desirable limit, while decrease to 7 per
cent, gives a great improvement, but
such thin blades tend to flutter. Then
in any case, but especially to permit
increase of thickness, mathematical
profiles of very small camber and
minimum ' maximum velocity ratio '
(Chapter VI) should be employed ; thus
a 10 per cent. Joukowski section has
been found * to give as good results as
an 8 per cent, section having a flat
under-surface. Finally, solidity may be
greater than for an airscrew working at
lower speeds. The problem becomes
most urgent, of course, in the case of
fast aeroplanes at high altitudes when
clearly rotational speeds should be low
and pitch large. FlG, 173A.

05 M at tip

242. Note on Preliminary Design

Systematic test results exist for families of airscrews of different
P/D ratios from which may be obtained by interpolation approxi-
mate figures for a projected design. Investigation on the lines
described above then determines whether the family shape can be
varied advantageously in view of special conditions, but these may
be so exceptional as to demand complete departure. The education
of an Aeronautical Engineer commonly includes the working out of

* Douglas. Aircraft Engineering, March 1936.

444 AERODYNAMICS [CH.

an airscrew from first principles. The exercise loses part of its
engineering value unless an attempt is made to fulfil a prescribed
specification, and, to avoid disappointment, approximate analysis
should precede detailed work on a proposed design.

Diameter. — Article 231 suggests a large diameter for efficiency, but
limitation arises as follows : (a) Aero engines turn at a fast rate, and
though gearing is used, its ratio is limited by weight economy ; thus
airscrews have a fairly high angular velocity, and diameter must be
restricted to avoid the velocity of sound (low temperature cuts down
this velocity, so that restriction is more severe on high-flying craft) ;
(b) long narrow blades of sufficient strength and stiffness have a dis-
advantage in weight ; (c) with small low-wing monoplanes having
outrigged engines, D may be limited by ground clearance ; in sea-
going craft large clearance is necessary on account of waves. A
formula in general use, as likely to lead to good performance in the
various duties required of a fixed-pitch 2-bladed airscrew, is —

D = 43 ( — -~

This diameter may be decreased by 10 per cent, for three and by 15
per cent, for four blades.

Approximate Allowance for Inflow. — An empirical method often
employed to allow roughly for inflow is as follows. An inflow
factor a is calculated from the formula :

2a) = -

1-2 pFaS

(where S is the disc area) and is assumed to be constant from the tip
circle to r = JZ), and then to fall linearly to zero at r = \D. T is
determined from the b.h.p. of the engine by interpolating from
systematic tests an efficiency appropriate to the chosen values of J
and P/D ratio. Rotational interference is ignored.

Shape. — To minimise torsional stresses, strains and oscillations,
the plan form of the blade is often made symmetrical about a radial
line joining the axis of rotation to the tip (cf. Fig. 167). This leaves
open the question of forward or backward tilt from the plane of
revolution. Forward tilt relieves radial stress, but as a rule is
resorted to only as a palliative, the blade being set parallel to the
disc. Highly loaded blades deflect forward appreciably, but this
usually makes for safety.

X] AIRSCREWS AND THE AUTOGYRO 445

Although flat-backed airscrews still occur, transverse sections
should be designed from a suitable mathematical formula to keep y
and compressibility effects small. If they are transformed shapes
(Chapter VI), incidences for zero lift will be known. It is essential
from practical and Aerodynamic points of view that the blade be
smooth and non-wavy as a whole as well as locally. In a particular
design this is verified by plotting geometric contours over the blade ;
these should be smooth curves and show absence of flats and con-
cavities. Eventual contours so disappointing as to necessitate re-
design are insured against by relating sections and incidences along
the blade in some systematic manner. Sections should be as thin

as is consistent with stiff-

0-032 1

0-024

0-016

0-008

ness, especially towards
the tips, but unavoidable
thickness close to the
boss makes it doubtful
whether attempts to
obtain thrust there are
worth while. Fig. 174
gives the minimum CD
at R = 10* for mathe-
matically designed sym-
metrical sections at zero
incidence through a range
of thickness ratio, maxi-
mum thickness occurring
at one-third of the chord
from the leading edge
(a considerably farther back position is preferable).

Stresses. — Radial stresses in a blade result from (a) bending due to
Aerodynamic force, (b) centrifugal force, (c) bending due to centrifugal
force. These are calculated separately for a number of radii and
added together (taking the fibre stresses, of course, from the bend-
ing). A first approximation to (a) is found by neglecting twist and
integrating, from the radius r0 considered to the tip, the element
moment CL%pW*c$r(r — r0). The least second moment of area
of the section is used to determine the fibre stresses, (b) is
self-evident, (c) arises at rQ only if the centroids of sections
nearer the tip are displaced from a radial line parallel to the
disc of revolution. The moment is resolved into components
about the major and minor axes of the section, and only the first
is considered.

0-1 0-2 0-3

Thickness Ratio

FIG. 174.

446 AERODYNAMICS [CH.

APPLICATION TO THE AUTOGYRO

243. A helicopter is a craft whose lift is derived from one or more
engine-driven airscrews with axes approximately vertical. The idea,
like that of flapping wings, has always attracted considerable notice.
Lift is said to be direct, since the craft can rise vertically. To
produce forward motion the airscrew axis may be tilted slightly
from the vertical. Evidently an advancing blade will then have
greater lift than a retreating one, rolling the craft unless combated.
Helicopters may have two concentric airscrews revolving in opposite
directions. Alternatively, roll could be avoided by increasing
incidence along retreating blades and decreasing that of advancing
blades. A method of achieving the same end, that is slightly
inferior Aerodynamically but much simpler mechanically, is to
allow the blades to flap up and down about axes near their roots as
they go round ; then the blade of greater lift reduces its incidence
by a vertically upward velocity component until its rise is checked
by a centrifugal moment about the flapping axis adjustable by stops
and springs. Single-rotor helicopters may have a small anti-spin
airscrew at the tail of the fuselage or an exhaust-gas jet.

The lift of an autogyro is derived from a windmill whose axis is
inclined backward from the normal to the direction of motion. In
flight, the rotor is not driven by the engine, which propels the craft
in the usual way, but automatically rotates by virtue of the relative
wind caused by forward or downward motion. Tilt of the axis may
be controlled from the cockpit. The blades of the windmill have
very small pitch, and they flap as above described. In the ' jump-
ing ' autogyro, pitch can be reduced and the engine connected to the
rotor ; high rotational speed is obtained on the ground in this way
and, on suddenly increasing pitch and returning the engine to its
normal duty, the craft leaps off, at first nearly vertically.

The autogyro has reached a practical stage of development in the
hands of its inventor, the late J. de la Cierva. Remarks on perfor-
mance appear in the next chapter. Meanwhile, we investigate its
rotor lift and drag, neglecting flapping in the first instance, since the
analysis is somewhat complicated.

244. Fixed-blade Inclined Windmill

In the theory * of the autogyro the lift Z and drag X are to be
obtained from the axial thrust T and a component of Aerodynamic

* Glauert, A.R.C.R. A M., 1111, 1926; Lock, A.R.C.R. <& M., 1127, 1927.

AIRSCREWS AND THE AUTOGYRO

447

force H which acts in the plane of the rotor away from the wind.
These are determined by the necessary condition for autorotation :
the torque Q = 0. The following usual nomenclature is adopted :

R = radius of tip circle and 5 = nR* ;
i = incidence of disc to relative wind (of velocity V) ;
Q = angular velocity of blades ;
X = 7/Q/e ;
[A = U/Q.R, where u = axial velocity through disc (assumed

constant) ;
6, <£= blade and helical

angles as before ;
fy = angle between blade
and direction of
motion viewed in
plan ;
T or Hc = T or

k, or kx = Z or X/pF'S ; C, or
Cx = Z or

In Fig. 175 (a), the broken line
represents the inclined windmill
in side view, and V is the resultant
of u and Fcos i as shown, u being
assumed constant over the disc,
It should be noted that u is in the
direction of T as the wind is
driving the rotor. If v is the
axial velocity induced by the
thrust, this is in the opposite
direction, and

v = V sin * — u. FlG- m-

Difficulty arises on account of the deflection of the slipstream, but,
on analogy with airscrews and aerofoils, it seems plausible to
determine v by —

T = 2pSvV = 2pS(7 sin i — u)\f(u* + V* cos1 i).
This gives —

i • (i)

X sin i =

+ X» cos1 t)

448 AERODYNAMICS [CH.

which is conveniently regarded as an equation for i, requiring know-
ledge of Tc and JJL.

As before, let W be the velocity normal to the blade of the element
at radius r (a radial component is neglected). From Fig. 175 (i),
which is a plan view in the direction of the axis, W is given approxi-
mately, on neglecting u* in comparison with W2, by —

W = £lr + V cos i sin fy. . . (ii)

For a single element, of chord

^ = rpW'cq . (iii)
'

where / and q have the meanings defined by (341). Now an auto-
gyro rotor is of very fine pitch and, though untrue at small radii, </> is
a small angle over important parts of the blade. Treating it as small
throughout gives the approximations :

2t = CL, 2q - CL<f> + CD.

As a further simplification we define 6 as for zero lift of the element
and take dCJdv. (a being its incidence) as 6 (cf. Chapter VI). Then,
since a = 6 —

t = 3(0 - $, q = 3<£(0 - $ + JCD.
Hence the second of equations (iii) reduces to —

Now <t> can be eliminated by the relation <£ = — w/W which follows
from <^ being small, and, remembering (ii) and the definitions of X
and [A, the torque grading along one blade becomes —

dO'

-JL- = ypcQ2 [—3|jLl?e(r + R\ cos i sin ^) — 3[x2^a

COS ^* Sil1 ^ + ^'^ COS*

No great error is introduced by assuming c and 6 constant along
the blade. Then integrating with respect to r gives for one whole
blade in a particular rotational position ^ :

Q' = pcn27?4[— (i0(l + |X cos t sin ^) — f (JL«

+ iCD (i + f X cos i sin <|j + iX« cos8 i sin' <!/)].

Now as the blade swings round and 4» varies through ± TC, clearly
the terms with sin ^ as factor will give zero mean torque, but the

X] AIRSCREWS AND THE AUTOGYRO 449

term with sin* <|* as factor will give a mean torque represented by
factor |. Thus for the whole rotor of B blades in rotation —

Q = B9c&R*[- ^6 - f (*« + JCD (1 + X* cos* i)]
and for a solidity or = BcR/S —

cos*

T = cr(8 + f (i + |6X* cos* 0

(352)
(353)

as approximations.
Similarly —

i*(CD -

(354)

The essential con-
dition Qc = 0 gives
the following equa-
tion for a :

O1-'

40°

(355)

Thus the problem
is determinate by
the tabular method
of step-by-step cal-
culation familiar in
airscrew work. kz
and kx follow from
Fig. 175 (a).

245. The modifica-
tion due to flapping
is left to further
reading. Glauert
shows first that a
trimming of 0 suffi-
cient to avoid rolling
means increase in Hc for an autogyro (though this is not true of a
helicopter), and that the alternative flapping must be rather worse.
The analysis indicates on this account rather better performance
than is to be expected in practice. Typical curves are shown in
Fig. 176 indicating that maximum CL occurs at a little short of 40°
incidence while the drag, at first low, equals the lift at 45°. Experi-

A.D 15

FIG. 176. — AUTOGYRO CURVES.
These are typical only and do not refer to a par-
ticular rotor. The broken line represents a complete
craft.

(NOTE : A, - JC.f kt = JCf.)

450 AERODYNAMICS [CH. X

mental results of reliability are scarce because scale effect is great
under wind-tunnel conditions. Some tests in America * on 10-ft.
diameter rotors in a 20-ft. wind tunnel indicated a maximum L/D of
about 6. Maximum LjD occurs at a very small incidence corres-
ponding practically to top speed of the craft, in contrast to an aero-
plane ; modern trend seeks to augment its value by reduction of
solidity, which is already considerably smaller than for an airscrew.
Decrease of solidity is compensated for by greater rotational speeds
and pitch. Limitations arise through centrifugal stresses becoming
great with the former variation while, in regard to the latter, pitch
must remain comparatively small to secure autorotation at all.

* Wheatley and Bioletti, N.A.C.A T.R., 552, 1936.

Chapter XI
PERFQRMANCE AND EFFICIENCY

246. The General Problem

The subject of performance, dealing with steady motion character-
istics of particular aircraft, has been introduced already at some
length in Chapter IV. We are now in a position, however, to
estimate data there assumed, and to criticise results. Critical
appreciation of values helps the designer to frame a suitable craft
for a specific duty, to verify that it will be successful, and to detect
and locate by full-scale trials minor deficiencies in the craft built,
whence engineering development follows. In the present chapter
aircraft are regarded essentially as competitive engineering products,
so that improvement by 1 per cent, is worth considerable endeavour.

The resistance of a complete craft when its airscrews are working
at their experimental mean pitch, i.e. giving zero thrust, is known as
its glider drag, and the first step is to assess this correctly. We
may proceed from model tests and aerofoil, airscrew, and skin
friction theory, calculating such scale effects as we can, and estimat-
ing others from experiment and experience. This method is
essentially one of summation, approximate allowance being made
by calculation or in experiment for the velocity field in which each
part works. In the case of a new type no other course is open
unless a compressed-air or giant tunnel is available for tests. But
with a more normal aircraft we can make much greater use of
experience, directing theory and experiment to assessing differences
from a nearly similar craft of known performance.

Much organised experimental work has been carried out in aid of
the first method, of which a brief outline is given in following
articles. Uncertainty arises chiefly from scale effects on the form
drags of separate component parts and the interference between one
part and another, over and above that which can be allowed for by
taking velocity fields into account. Form drag and interference,
apart from airscrew effects, may readily absorb 26 per cent, of the
power at top speed, so that resulting errors may be appreciable.

451

452 AERODYNAMICS [CH.

Although interference usually increases drag, this is not always
the case. Exceptions include the Handley Page wing slot at large
incidences and the Townend * ring for radial engines (cf. a paper by
Otten f and several others { on full-scale tests).

The second method of assessing glider drag is largely individual
to the designer, being founded upon experience with different types.
Fairly similar aeroplanes may be grouped and each category assigned
a gross drag coefficient, which is often based on the frontally pro-
jected, or so-called ' flat plate/ area. Particular idiosyncrasies are
met by adjusting the coefficient.

With glider drag known, the performance of an aeroplane in a
standard atmosphere is readily forecast by taking due account of its
airscrews and engines. Important interference occurs between
airscrews and fuselages, or engine nacelles, a matter that is studied
in a specialsection. Assuming reciprocating engines, their rotational
speed becomes a significant variable.

The advent of jet propulsion promises some simplification of the
present calculations and also justifies consideration of performance
and operational efficiency from a comparatively unhampered point
of view.

To be representatively intelligible, particular flight tests must be
reduced to standard atmospheric conditions and the chapter con-
cludes with a note on a suitable procedure to adopt.

COMPUTATION OF GLIDER DRAG

247. The experimental drag coefficient of wings, tail planes, and
other aerofoil surfaces is usually written —

CD = Cpf -f- CDO>

the first term on the right being the induced and the second the
profile drag coefficient. This form is suitable when tests have been
made at flight Reynolds numbers, e.g. when C.A.T. measurements
are available and the craft is small and slow. Except in such
circumstances, however, a more useful subdivision is —

CD = Cm + CD (friction) + CD (form) . . (356)

The advantage and purpose of (356) is to isolate the part of the drag
— the skin friction — whose scale effect is easily predicted. This
term can be obtained by subtracting from the gross drag an experi-

* A.R.C.R. & M., 1267, 1929 ; Aircraft Engineering, April 1930.

t Jour. Roy. Aero. Soc., November 1934.

t Cf. Van der Maas, 6th Int. Con., The Hague, 1930, etc.

XI] PERFORMANCE AND EFFICIENCY 453

mental form drag (Article 230), but when, as is usual, the data do
not exist, an estimate can be made of one or other of the two com-
ponents of profile drag. It will be assumed that only the friction is
subject to increase by roughness.

The formula is not confined to aerofoil surfaces, although for
others the first term will probably be negligible. With bluff com-
ponents such as wheels, the last term swamps the others, and an
experimental CD is used and scale effect is neglected except in cases
where it has been especially determined. Each exposed component
of the craft will first be assigned a drag coefficient based on an area
peculiar to itself, but in the summation a common area is more
convenient. For the purpose of comparing with other aircraft or
checking against experience the flat plate area of the preceding
article may be used, but the area finally chosen will be the wing area.
Usually this area is taken to include parts of the body and engine
nacelles intercepted between the leading and trailing edges.

Airscrew effects are reserved for the next section.

248. Induced Drag

Induced drag can be neglected except on the wings, tail plane, and
fin-rudder unit, although with a very large body this may not be
justified. Its calculation is treated in Chapter VIII, but the follow-
ing additional remarks are made.

It is convenient, in the case of a biplane, to determine the equival-
ent monoplane aspect ratio, the equivalent monoplane being defined
as having the same lift and induced drag as the biplane. Denote
lift and drag by L and D, span by 2s, and L/2s by X ; use suffix M
to distinguish the equivalent monoplane, and 1 the longer and 2
the shorter of the biplane wings ; and write JJL for st/Si. Now
Dm = DiEt the induced drag of the whole biplane. Hence from
(251) and (252)—

X£ = V + X.1 + 2orX1Xt . . (i)

a being Prandtl's factor. Let L^ = xLH. Then, since L^ + £1 =
Z,M, LtjLl = (1 — x)/x. Dividing (i) through by X^ and substitut-
ing—

or —

i* = - <S{3P(\* - 2a[x + 1) + 2#(<ifjL - 1) + 1} . (357)

% \L

Writing the R.H.S. as l[K, 2Ksl = 2sM is the span required.

454

AERODYNAMICS

[Cn.

This result can be plotted in several useful ways with the help of
Fig. 135. One of these is shown in Fig. 177 which exhibits, amongst

other matters, that
the biplane is most
efficient when s2/s1
= £,/!,! = 1 (Arti-
cles 183, 185).

Once the distribu-
tion of lift between
the wings has been
estimated by direct
experiment or other-
wise,* the theoreti-
cal induced drag
follows immediately
from Chapter VIII.
Partly for reasons
discussed in Article
180, however, and

FIG. 177. — EQUIVALENT MONOPLANE ASPECT RATIO. partly to allow for

irregularities of lift

distribution arising from practical causes, the drag so calculated
is regarded as a lower estimate and up to 8 percent, added. This
remark applies equally to monoplanes. Of course, should the theory
be used only to assess differences between two rather similar wing
arrangements, the correction would not be needed.

The induced drag of a tail plane is assessed from the ± lift it must
exert to preserve equilibrium. Induced drag arises particularly on
the fin and rudder when the craft is flying yawed owing to partial
engine failure, a first approximation to the crosswind force being that
required to balance the moment about the C.G. of the asymmetrical
thrust. A small increment is due, however, to the fin being set
slightly askew to balance the torque of the airscrews, which revolve
all to the same hand, unless contra-propellers are used.

249. Form Drag

Chapter IX A provides data for directly estimating the profile
drag of smooth normal wings at small incidences and of laminar flow
wings except through the favourable range. Within this range the
skin friction may be calculated approximately for laminar flow
wings but the question arises as to what form drag to add, a matter on

* Diehl, N.A.C.A.T.R/s, 468 and 501

XI]

PERFORMANCE AND EFFICIENCY

455

0 Q-2c 04C 0 02c 04c

DISTANCE OF TRANSITION POINT FROM NOSE

FIG. 178.— -THICKNESS: (a) 0-14, (b) 0-25. R: 107, 5 X 107.

which the experimental data generally available are not yet sufficient.
Special tests can easily be made in a laminar flow tunnel, the two-
dimensional conditions required enabling good Reynolds numbers
to be reached, but few of these tunnels exist. Fig. 178 summarises
results of Squire and Young's calculations of the form drag of normal
wings, which may be used for laminar flow wings provided maximum
thicknesses are only moderately displaced backward ; evidently the
curves cannot be extrapolated indefinitely owing to increasing
bluntness of tail.

These data can also be applied to tail-unit sections with the same
restrictions, but allowances necessary for aileron, elevator and
rudder angles, excrescences and roughness are commonly consider-
able and should be determined by test.

What has been said of aerofoil surfaces applies in principle to
fuselages.* With these a tunnel test is usual, but form drag is
difficult to infer. If the shape is a very good one and free of ex-
crescences, a repeat test may be carried out after fairing in the wind
screen, which commonly has a large form drag, and the whole drag
then measured will be an upper limit to the skin friction. Form
drag is otherwise caused by wing roots, the downwash field, the tail
unit, and, in case of engine failure, by yaw (postponing discussion of
slipstream effects). Failing C.A.T. or giant tunnel tests, estimates
may be based on measurements with two models, a small and a very
large one. The first is tested principally for pitch and yaw in-
creases ; the second only at zero incidence. Wing roots, tail
surfaces, and a model tail wheel may be offered up in the second case
to give some idea of the interference drag on the body.

It will be realised that wings, tail unit, and fuselage all have

* Some recent tests of streamline bodies of revolution at Reynolds numbers
(reckoned on length) between 2 and 3 X 106 gave for the ratio of form drag to skin
friction : 0-08 with streamline, and 0-14 with turbulent oncoming flow.

456

AERODYNAMICS

[CH.

different Reynolds numbers on the craft. It is profitable and entails
little complication to take this into account.

250. Parasite Drag

The term parasitic is usually applied to the drag of all parts of an
aircraft except its lifting surfaces. Thus the inter-plane bracing of bi-
planes, aileron control horns, tail planes, etc., contribute to parasite
drag, but wings, lifting rotors, and gas envelopes do not, though part

TABLE IX

Component

Drag at optimum
incidence referred to 100 m.p h.

Fuselages :

Best, high speed .....

Exceptionally good ....

Average ......

Square section with protuberances .
Flying-boat hull, best ....
Seaplane floats, good shapes
Biplane bracing :

Single bay, engines in body or wings

Ditto, without cabane panels .

Two bay, with engines between wings
Tail plane, fin, and rudder :

Slow, inefficient craft ....

High-speed craft .....
Tail-unit hinge breaks ....
Shrouded aileron slot ....

Tail skids, small craft ....
Low-pressure tail wheel and scantlings for

medium-size craft.
Medium-pressure wheels, faired rim to hub .

Non -retractable undercarriage : .

Single-engined craft. ....

Twin-engined biplane ....
Small craft, cantilever ....
,, ,, tripod ....

,, ,, inferior ....
External oil-cooler, medium-size engine
Exhaust pipes, ditto engine
Radial engine cooling drag with ordinary
cowling.

2J Ib. per sq. ft. maximum sec-

tional area across stream.

3 Ditto

4£ Ditto

7 Ditto

*2J~4 Ditto

4-6 Ditto

0-1 P, including interference.
0-07 P, Ditto

0-16 P, Ditto

0-07-0-1 P, Ditto

0-2 P, Ditto

Add 15 per cent, to min. drag.

Add 12 per cent, to min. drag of

wing span affected.
1-4 Ib.
20 Ib. (small craft, 6 Ib.)

11-14 Ib. per sq. ft. of projected
area of tyre.

0-18 P,

0-13 P,
15 Ib.
30 Ib.
40 Ib.

including interferences,
which often contribute
16 per cent, of whole.
Ditto

10 Ib.

Equivalent to 7 per cent, of the
b.h.p. of the engine. (Note. —
Such losses are greatly reduced
with ducted cooling.)

* Remarkably low drag of this order appears to be realised in some recent flying
boats by (a) very careful shaping, (6) reducing the beam of the hull while lengthening
the forebody to maintain planing surface (Gouge, lecture before the Roy. Aero. Soc.,
December 1936).

XI] PERFORMANCE AND EFFICIENCY 457

of the resistance be form drag and perhaps reducible. Interference
drag is reckoned parasitic if due to struts, fuselages, engine nacelles,
etc., even though it appear as an increase in the form drag of lifting
members, but it is excluded if arising between lifting surfaces, e.g.
the mutual interference of the wings of a biplane.

Reference must be made to Handbooks and Laboratory Reports
for design data. Table IX is intended to indicate merely the
beginnings of an adequate list of notes that should be prepared and
constantly revised to facilitate and check estimations. In this
table 100 m.p.h. is used only as a convenient reference, and it is not
implied that scale effect is to be allowed on either side of this speed.
Drags are sometimes expressed as fractions of the sum of parasite
drags, which is denoted by P. (The term total parasitic drag is
reserved to include the profile drag of wings.)

251. Struts and Streamline Wires

Curve (1) (Fig. 179) shows the variation of CD (= drag per unit
length -f- |p^*r, where T denotes the maximum thickness of the

010

O05

4-4 4-6

4-8* 50 52
log* (VT/V)

5-4 0° 2° 4° 6°

2 r. 4 6 Angle of Yaw
Fineness

FIG. 179. — DRAG OF STRUTS AND WIRES.

section) for struts of approximately minimum drag at Iog10 (FT/v) =
5*4. Optimum fineness is 4 ; larger Reynolds numbers permil a
slightly greater thickness, smaller Reynolds numbers require it to be

A.D. 15*

458 AERODYNAMICS [CH.

less. This minimum drag is plotted against R in curve (2). To
secure such small values the contour must be Aerodynamically
smooth and mathematically designed, although slight rounding may
be introduced at the trailing edge. Curve (3) relates to the strut
illustrated, which has a fineness ratio of 4, but whose section is
slightly thicker at the back. Variation of the drag of this strut with
yaw at Iog10 R approximately 4-7 is shown by curve (4). Although
the section figured does not accord with minimum drag, it is a very
good one ; carelessly designed sections are often 20 per cent,
wasteful.

The strut of minimum drag may not be quite the most efficient.
Its weight w must be supported by the wings, say of L\D ratio rt and
causes an additional drag w/r. If D is its drag, the criterion for
efficiency is that D + w/r be a minimum. The central cross-section
of the strut will be required to have a given minimum moment of
inertia ; weight is saved quickly by increasing thickness for given
strength, but the advantage is limited with r large. Note that any
question of changing the struts on an existing craft would be differ-
ently decided, because, wing area being fixed, r would then be the
over-all L/D ratio, the craft having to fly slightly faster to carry
additional weight at the same incidence. Such calculations some-
times indicate, for a large, slow biplane, rather a small fineness ratio,
which, however, is to be avoided, for a curious reason : the cross-
wind force arising on the struts in yaw becomes uncertain in direc-
tion, i.e. it may act in the direction of the sideslip, and the uncer-
tainty slightly affects lateral stability. Considerable saving results
from tapering struts.

Wires of the lenticular section also illustrated in the figure give
curve (5). Their drag at the smaller Reynolds numbers of usual
interest is given by :

logwl? . . .4-1 4-2 4-3 4-4 4-5 4-6

CD X 10 . . . 3-00 2-58 2-14 1-72 1-34 1-02

R and CD being defined as for struts. Lenticular wires show little
increase of drag for yaw of their sections up to 10°. With double
wires a short distance apart, one directly behind the other, shielding
to the extent of 15 per cent, may be allowed, but interference
increases the drag of each with yaw > 8°.

A rough rule to allow for interference drag in the bracing of a
biplane is to add 50 per cent, to the drag calculated for the interplane
struts and wires.

XI] PERFORMANCE AND EFFICIENCY 459

252. The Jones Efficiency

Analysis of a number of first-class monoplanes of exceptionally
fine lines and smooth surface, varying in all-up weight from 2 to 8
tons, gives the mean results of Table X. Single and twin engines are
included. Top speeds vary from 200 to 250 m.p.h. ; the Table
applies to 220 m.p.h. with a wing loading of 21 Ib. per sq. ft. and a
' power loading ' of 12-3 Ib. per b.h.p. Drags are expressed in
round-number percentages of total drag. The large allowance under
the last item includes a non-retracted tail wheel or skid, non-ducted
cooling of reciprocating engines, and airscrew interference.

TABLE x

Component

Nature of Drag

Totals

Induced

Friction

Rough-
ness

Form and
interfer-
ence

Wings and ailerons .
Tail plane, fin, and rudder .
Fuselage ....
Engines and miscellaneous.

Totals

7
Small
0
0

31
8
14
4

5
1
2

Small

8
3
3
14

51
12
19
18

100

Points to notice in these small and lightly loaded craft are : that
one-half of the thrust h.p. is expended on the wings, in spite of the
fair speed ; that good shaping and smoothness make pure skin
friction account for more than one-half power ; and that induced drag
is very small at top speed (it would be substantially increased, of
course, at cruising speed or with a heavier wing-loading).

Various different meanings are commonly attached to the term
efficiency. One, which will be distinguished as the Jones efficiency,
depends on the valuable conception of the streamline aeroplane* and
is defined as follows :

where Hi = horsepower absorbed by induced drag.

HF = horsepower absorbed by pure skin friction.

Y] = efficiency of the airscrews.

It will be seen that lack of an allowance for form drag tends to
low values, the present idealisation leaving only induced drag and
pure skin friction. Usually the convention is to calculate the skin
friction from flat plate theory, whilst, again, the induced drag is

* Jones (Prof. Sir Melvill), Jour. Roy. Aero. Soc., May 1929.

460 AERODYNAMICS [CH.

assessed from the theory of Chapter VIII. The resulting figure of
merit is reduced on both these scores through lack of sufficient
knowledge. However, this figure is easily calculated, and does
succeed in segregating good classes of aircraft from bad, and penal-
ties incurred by excessive form drag, slipstream effects, and rough-
ness are made plainly evident.

From Table X we at once find, since (358) is the same as the ratio
of the sum of induced drag and pure skin friction to total drag, that
the monoplanes investigated have the mean efficiency of 64 per cent.
Aeroplanes having an efficiency > 60 per cent, on this basis are
considered satisfactory at the present time ; the slow, multi-strutted
biplanes of prior to 1930 often had an efficiency of only 30 per cent.
Considering the possibility of substantial improvement of efficiency,
we note that the following steps promise immediate profit : (a)
elimination of roughness drag ; (b) reduction to a very small mini-
mum of eradicable parasite drags (expressively called Christmas-
tree drags), as by retracting the tail wheel and closing all slots when
not in use ; (c) reducing engine losses by ducted cooling ; (d) mathe-
matical design of every contour to reduce form drag to a minimum ;
(e) study of the manner of joining one part to another to decrease
interference increments ; and finally (/), in the case of craft of small
speed range, designing wings to approximate more closely to elliptic
span distribution of lift. All these steps are in progress, and the
present best efficiency of 65 per cent, need by no means be considered
the maximum that can be attained.

The above method can be applied to airships, and best examples
show a slightly higher efficiency, but, of course, at so low a speed
that comparison with aeroplanes is not justifiable.

253. Subdivision of Parasite Drag

The number of flight conditions at which drag must be estimated
are few. Normally they are : top speed, cruising, climb at low
altitude, ceiling, ceiling with engine failure, coming-in incidence,
nose dive or fast glide. With an efficient aircraft it is best to estimate
for these conditions separately, since variation of Reynolds numbers
and various small corrections are then usefully taken into account.
When a large number of rather similar craft are dealt with, however,
or when the type is inefficient, comprising a host of parasitic resis-
tances whose scale variations are quite unknown, labour is saved by
assembling the drags in two groups : those dependent on incidence
and those which are not. Analysis shows that variation of the first
group, which includes the fuselage and tail plane, may usually be

XI] PERFORMANCE AND EFFICIENCY 461

expressed as a function of ACJA^Ci., the difference in lift co-
efficient being reckoned from that for top speed. The form of this
function depends on the type of craft, and, unless directly calculated
for a given type, must be suggested from experience. Examples are
given by Kerber.*

AIRSCREW INTERFERENCE

254. In Chapter X the airscrew was regarded as isolated. A large
engine nacelle or body in front of or behind the airscrew modifies its
torque and thrust, while the drag of the body is increased by the
acceleration of air through the disc. Theory and experiment show
that the pitch of the airscrew and therefore its efficiency are appar-
ently increased owing to the slowing up of the stream by the body.
If, however, the increase of drag of the body be subtracted from the
apparent thrust, to give a useful thrust, the effective efficiency is
found to be less than for the isolated airscrew, unless the spinner
appreciably improves a bad body shape. It will be seen that the
mutual interferences are to some extent compensating, and in some
flying-boat arrangements particularly, with engines carried high in
separate eggs, it may be convenient to deal with an over-all correc-
tion. But more frequently we have to determine with some care
increments of wing and parasite drag additional to effects on the
engine housing.

Detailed investigation in practical circumstances, though possible,
is very intricate, but certain simple factors have outstanding impor-
tance, and consideration of these alone is usually sufficient for
performance calculations. However, unless the airscrew is designed
to conform with the actual velocity field in which it works, modifica-
tion of blade angles will be necessary to allow the engine to develop
full power. We first neglect all aircraft parts other than the nacelle
or body. Experiment shows that the form of the results obtained
in these simplified circumstances is retained with the complete craft.
Alternatively, if experimental determination of coefficients is not
available, we shall be able to supplement the results by additional
calculations.

Whether the body is behind or in front, some shielding takes place,
either of the nacelle by the boss and blade roots in the former case or
vice versa in the latter. Thus, usually drag is decreased on this
score by a small fraction, which, we shall denote by kb. The special
case where the boss is flush with the body, of whose shape the spinner
forms an integral part, will be referred to later.

* Aerodynamic Theory, v. V, 1935.

462 AERODYNAMICS [CH.

Again, with both tractor and pusher type airscrews, the body is
situated in a field of augmented velocity, increased by from aV to
2aV in the first case and from 0 to a V in the second, a being the axial
inflow factor. We shall associate the resulting increase of drag
with a certain coefficient kv. This acceleration is accompanied by
pressure variations which again increase drag.

Finally, in the case of a tractor airscrew, the sudden increase of
pressure at the back of the disc, by virtue of which thrust arises,
produces important mutual effects. Pressure drag increases will be
connected with a coefficient kp.

255. Tractor Arrangement

It is convenient to begin with the pressure drag last mentioned.
If the flow were inviscid, a small streamline body close behind the
airscrew would experience this drag in full, yet clearly the efficiency
of the combination would be the same as that of the isolated airscrew,
for no energy loss could be caused by the body, and the residual slip-
stream would be unchanged by it. We infer that the thrust of the
airscrew must increase to compensate exactly for the pressure drag
introduced.* Experiment bears out the practical value of this
conclusion. Displacing an airscrew upstream from a body, as if it
were driven through an extension shaft, decreases apparent thrust,
originally high owing to interference, but also decreases the drag of
the body equally, so that effective or net thrust remains unchanged.

Let Ta be the apparent thrust of the airscrew and Se its effective
disc area (Article 240) ; Da, D the drag of the body with and without
the airscrew, and S its maximum cross-sectional area. It is assumed
a sufficient approximation for present purposes to regard Ta as
uniformly distributed over Set so that pressure is proportional to
Ta/Se. Considering a proportion of S, depending on the shape of
the nose of the body and the position of the airscrew, to be affected
by this pressure, the interference thrust and drag increments can be
expressed as equal to Ta . kp(S/Se), and on this score only —

T* = I _ kp(S/Se) * ' • (359)
Da-D=Ta.kp(S/Sf). . . (i)

* The argument may be illustrated by an analogy. Supposing a plumb-bob to
hang from a spring balance, the decrease produced in its indicated weight on immersing
it in a beaker of water is equal to the apparent increase in weight of the beaker of
water due to the rise of water level. The upthrust on the plumb-bob is analogous
to the increase of propeller thrust in the original case, and the increase of hydro-
static pressure integrated over the base of the beaker to the increase in drag of the
streamline body.

XI] PERFORMANCE AND EFFICIENCY 463

Next, if the body were effectively wholly exposed to the ultimate
slipstream velocity Vs and S were sufficiently small compared with
S,, we should have on this particular score —

Da _

But—

Ta « PF (1 + a) 5, . 2aV
or —

2a + 2a* = Ta/9V*Se.
Hence :

Da/D = (1 + 2a)« - 1 + 2TJ9V*Se . (ii)

To take into account throttling of the slipstream by the appreciable
section of the body and other neglected factors, we write this :

>.... (iii)

Finally, collecting from (i) and (iii), and remembering that
shielding reduces drag by kbD, we have —

= D(l-kb) + 2v (kt + #,£„) . . (360)

where CD for the body is specified on S. Alternatively —

- . . (36,)

say.

256. Pusher Airscrew

The case of an airscrew working behind a body differs only in
detail. The body is subject to less increase of speed, which is con-
fined to its back part, but to a fully effective pressure gradient. On
the other hand, absence of a propelled body behind means that in-
crease of thrust is no longer compensated for by increased pressure
drag. These changes promise advantage to the pusher arrangement,
but this tends to be lost in practice by poor streamline shape for the
body and by necessary adjustment of airscrew form to ensure
realising the full b.h.p. of the engine. It appears from experiment

Da
0

464 AERODYNAMICS [CH.

that (362) still applies, with suitable adjustment of the coefficients,
so that special investigation is unnecessary except to note that B
will be smaller owing to the relative unimportance of kv.

257. Comparison with Experiment

The simple linear relationship (362) (see Fig. 180) has been realised
experimentally on many occasions during the past thirty years,

2 and it has been found to hold

for complete aeroplanes as
well as in the simplified cir-
cumstances assumed for its
derivation. Definition of the
coefficients is slightly modified
in experimental work, so that
the actual airscrew disc area
can be used in place of its
effective area. With this un-
derstanding, tests on a biplane
of very poor Aerodynamic
shape gave, without the wings
in position, A = 0-86, B =
1-04, and with the wings, A =
0-83, B = 0-93. In a more
recent analysis kv was found
to be 2-4 (rather greater, as expected, than 2, as in (ii), Article 255)
and kp = J, so that with CD = 0-2, for example, B would be 4-9.
Values are not yet systematised.

Whilst A is found to be < 1 for blunt-nosed nacelles as predicted,
this no longer holds when the boss and spinner complete the lines of
the streamline housing of, say, a liquid-cooled engine. kb may then
vanish ; or it may change sign, and considerable positive drag then
result, with fine bodies, owing to deterioration of flow.

Some pusher naceUes have given A = 0-97, B = 0-83 for a poor
shape, and A = 1-05, B = 1-8 for a somewhat better one.

258. With high-speed monoplanes having two or four radial
engines, the nacelles project for efficiency about one-quarter chord
in front of the leading edge of the wings. Considering one of the
inner engines, we have behind its airscrew a long nacelle followed by
a strip of wing of approximately maximum chord and, at a little
distance, possibly one-third of the tail plane area. With a single-
engined aeroplane, the slipstream affects, besides the fuselage, the
wing roots, tail plane, fin, and rudder and, if non-retracted, part of

Ta//>V2Se

FIG. 180.

XI] PERFORMANCE AND EFFICIENCY 466

the undercarriage, together with certain struts and wires in the case
of a biplane. Actually, the question of precisely what components
of the after-part of a craft are affected is somewhat doubtful, for
slipstreams are found to wander. Associated with this question is
that of what diameter to ascribe to a slipstream ; on the one hand,
it contracts appreciably, especially at maximum climb, when slip-
stream effects are most important ; on the other, some distension is
caused by the blanketing of the central part by a large bluff nacelle.
It is usually sufficient to assume that parts affected lie in the pro-
jection downstream of the effective disc area, and that the additional
velocity is 2aV spread uniformly across the slipstream. Alterna-
tively, the method indicated in Article 242 may be applied. Com-
parison is made in the following example :

Example. — We take an ordinary small biplane with a single engine
of 300 b.h.p. and a 9-ft. airscrew giving 240 effective thrust h.p.,
which provides a top speed of 150 m.p.h. (pFa =• 115). The maxi-
mum cross-sectional area of the body is 16 sq. ft., and its drag
without airscrew 166 lb., so that CD = 0*18. In order to use J for
kpt we substitute Sa for S, in (359) and, taking Sa = 64 sq. ft.,
obtain —

Thus thrust is increased by 40 lb. and the body drag by an equal
amount through pressure interference. (362) gives for the fuselage
on assuming A = 0-86 :

or the drag of the body alone increases by 51 lb. The glider drag of
the remainder of the craft at top speed is estimated to be 360 lb., of
which parts contributing 90 lb. are washed by the slipstream. By
(351) and particulars of the airscrew, the effective disc area is found
to be 41 sq. ft. From Article 255 (ii), (1 + 2a)« = 1-272, and the
90 lb. drag concerned becomes 114 lb. If, alternatively, we applied
the approximate method of Article 242, we should find —

a(l + 2«) = r/1'2 x 115 X 64

and since T = 240 x 550/220 = 600 this would give (1 + 20)» =
1-273, so that in the present instance little difference would result.
For the complete aircraft at top speed D =217 (body) + 270 (parts
out of slipstream) + 114 (other parts within slipstream) = 601 lb.,
and the total increase due to airscrew interference is (51 + 24)/(166

466 AERODYNAMICS [CH.

4. 360) = 14 per cent. The airscrew efficiency, provided the blade
angles can be adjusted, without further loss, to absorb 300 b.h.p. at
the designed engine speed, is increased from 240/300 = 80 per cent,
to 640 x 220/550 X 300 = 85 per cent., the additional thrust being
balanced by extra pressure drag. Much greater airscrew interfer-
ence would occur, of course, at maximum climb.

1-0

SOME PERFORMANCE CALCULATIONS

259. Prediction of Speed and Climb

The only development now introduced compared with Chapter IV
is in the account taken of engine and airscrew characteristics. The

following are assumed : a
' polar ' of CL ~ CD for the
complete aeroplane whose
performance is required, all
lifts and drags being referred
to the wing area S ; curves
of variation of &T = thrust/
pn2D4, and kQ = torque/ pwajDB
against J = V/nD for the
propeller of diameter D
working at n r.p.s., cor-
rected to position on the
craft ; and a maker's curve
(e.g. Fig. 181 (a) ) of H0 the
b.h.p. of the engine against
n at sea-level, often called
the standard b.h.p. Other
symbols are : L, W for total
lift and weight, T for thrust,
a for relative air density, and
v for rate of climb in feet
per second. Suffix 0 refers
to sea-level. To prepare for
change of altitude we sub-
stitute F-\/o-, n^a (cf.
Article 81) for V and w,
when —

08
Rli)
06

0-4

NORMALLY ASPIRATED/^

5 10 15 20
ALTITUDE IN THOUSANDS OF FT

FIG. 181. — VARIATION OF ENGINE POWER
WITH ALTITUDE.

The curve (a) is an example only of
variation of power with engine speed, and
must not be taken as representative.

r
C =

r\
(i)

XIJ PERFORMANCE AND EFFICIENCY 467

(ii)

and similarly for other coefficients. Equation (i) is assumed as
indicated to hold, with W written for L, as a sufficient approximation
in level and climbing flight.

For any altitude h we write for the b.h.p. available —

H=H0.f(h) ..... (363)

Different forms occur for f(ti). One formula in use for supercharged
engines is __

<364)

This curve is shown dotted together with another mean curve for
normally aspirated engines in Fig. 181. In the formula, p of course
denotes the pressure and T the absolute temperature of the atmo-
sphere. The torque coefficient is readily expressed in terms of
(363) :

(365)

ner
275 (366)

These last formulae depend only on the engine and airscrew, and
hence corresponding values of n and V are known, whatever CL or CD
may be ; knowing /(A) for a particular engine, we can construct for a
given airscrew a family of curves of V against n or /, one curve for
each altitude, which will represent the best that the engine and air-
screw can do.

Now, turning to the aircraft, unless the angle of climb is steep —

W . . . (367)

or in terms of coefficients * :

CL • • • (368)

Thus the next step is to plot a curve of k^/}* against /, which follows
immediately from the airscrew data.

Level Flight. — The formulae may be used as follows. Choose a
value of K\/cr and obtain CL from (i) and CD from the polar. Find
JfcT//a from (368) and then / from the data curve mentioned. Read

* Modification for more than one airscrew will be apparent.

1-6
1-2

0-8

U-Ulb

—

n-ni7

llAllC

0008

0-2
n

0004

CHM O08 0-12 (M6 O 2

POLAR FOR COMPLETE
CRAFT

OTHER DATA>-

TOTAL WEIGHT- 10 TONS.
ENGINES: 2-800 BHP.
1700 r.p.m. AT G.L.,

«/HMAX~ "/"MAX
AS IN FIG. 181 (a),

(NORMALLY ASPIRATED).

0-5 (W 07 0-8 0-9
J

AIRSCREW CURVES
(P/D-1-0)

S/2D2«5-0
DHIFT.

ENGINE CURVES:- SEA LEVEL

20,000 FT'

FIG. 182. — EXAMPLE OF PERFORMANCE PREDICTION.
fiT, gives the total power of the two engines.

468

PERFORMANCE AND EFFICIENCY

469

CH. xi]

off kQ from the curve kQ ~ /. Thence obtain H0 . f(h)]n from (366).
Repeat the process, and finally plot the last quantity against n VCT ;
the curve obtained is unique for the aircraft in level flight at all
altitudes ; an example is given in Fig. 182.

Thus we have a single curve of H0 -f(h)/n ~ n«/a for the aircraft
and airscrew.

Also, from the family of curves mentioned under (366) we can plot
a family of curves of the same quantities : H0 . f(h)/n & n^/a, for the
engine. Examples are

(r.p.m.)A/cr
900 1 100 1300 1500 1700

:200

300

i i fry

5.000 FT.
10,000 FT.

150

ct
E

marked in Fig. 182.

Intersections satisfy
the requirement : Hf\ =
thrust h.p. required, and
represent top speeds at
various altitudes. These
intersections may be
plotted in several ways.
Since corresponding
flight values of V and n
are now known, we can
find, for instance, a
unique curve of n^a ~
F\/cr, each point on it
representing a particu-
lar altitude ; an example
is given in Fig. 183,
where also the ceiling is
marked. It will be
noticed that n falls away
as h increases.

Climbing. — On assum-
ing a value of V9 the
corresponding value of
/ follows from ^365) or
(366), as already traced,
and thence k^. But the value of V gives definite values of CD and
CL. Hence v is calculated from (367) rewritten in the form :

200

IVcr

100

-CEILING 20.500 FT

FIG. 183.— PLOTTED FROM FIG. 182.
These details relate to flaps in . The minimum
flying speed with flaps out at low altitude is
60 m.p.h., approximately.

_ 2k
~ T

(369)

This calculation is, of course, appropriate to a particular altitude,
and assumption of other values of V enables maximum climb at that

470 AERODYNAMICS [CH.

altitude to be found by plotting. Repetition with other densities
gives the maximum rate of climb as a function of altitude, whence
time of climb is estimated as in Chapter IV.

There may sometimes be slight advantage in this analysis in
having V in m.p.h., n in r.p.m., or in using the geometrical pitch in
place of J.

260. Another arrangement * of the foregoing process will be
described briefly. Instead of k^ and &Q we employ Tc = T/pF2/)*
(cf. Article 232 and note that Tc = &T//2) and a b.h.p. coefficient
JcH = 550 #/pw8D6. Both these must be plotted against J for the
airscrew concerned. The new thrust coefficient can be written,
since TF/550 = TJ# or Tc pF8D2 = 7jfcH pw*D5, as—

Now for any assumed value of n we can find &HO with the help of
the maker's engine test sheet, and thence J from the curve of &HO ~ J,
finally obtaining V, so that repetition gives a curve of V ~ n for
sea-level. Different curves are required for other altitudes of
interest. To find these we write for altitude h :

and, knowing /(A), find for each altitude a curve of n\Sc ~ V\/a.
These curves define the revolutions available for the given engine
and airscrew.

Turning to the aircraft in level flight, and putting T = drag as a
sufficient approximation, clearly :

Tt=tC»^ . . . . (370)

A unique curve of n^/a ~ V VCT is determined by assuming first some
value of V \/G, which gives immediately CL, and thence from the polar
for the complete craft CD, and Tc from (370), then reading off / from
the curve Tc ~ / and calculating n<\/a from the obvious relation-
ship —

n^/a^V^/G/DJ. . . . (371)

Repeating this process gives the curve required, which holds for all
altitudes.

Intersections of the engine-airscrew curves with this aircraft curve
give values of V<\/G for various altitudes.

Climbing. — From (367) we obtain, in place of (369) (T>drag) —

• • - (372)

* The method of the preceding article is known as Bairstow's (cf. Applied Aero-
dynamics) and the present as the Lesley-Reid, N.A.C.A.T.N., 302, 1929.

XI] PERFORMANCE AND EFFICIENCY 471

So if we assume a value of V\/at and find the engine-airscrew «\/a
and thence / from (371), Tc will follow, whilst CL and CD are known,
because we are taking as a sufficient approximation that lift =
weight during climbing flight. A series of values of v/V so obtained
enables maximum v to be determined at the chosen altitude.

26oA. More elastic methods suffice to solve isolated questions
relating to the performance of an aeroplane driven by airscrews.
The data provided commonly comprise, besides the all-up weight
W — airscrews : the number, diameter (Z>), and &r, kQ at various
values of / ; engines : such particulars as enable AQ to be related
to n, the airscrew revolutions per second ; wings : the plan-form
together with the area (5), the span (2s), and possibly a polar curve
of CL plotted against CD ; extra-to-wing drag : an estimate of the
glider drag apart from the wings at some intermediate speed ; and
lastly, information as to the fraction of the total glider parasitic
drag (Dp) that is affected by slipstreams. The following discussion
refers for clearness to a single-engined aeroplane, and the effect of
the slipstream on the induced drag (/),-) is ignored.

The usual procedure is to work out a long table of calculations,
of which the first two rows comprise the given values of / and kQ,
so that each column of the table relates to a particular value of J.
In the absence of a constant-speed airscrew, row 3 lists the corre-
sponding values of n. To obtain these it is often possible to express
the engine data in the form b.h.p. = kn* for constant altitude, and
equating kn*to 2nn . &Qpw2D5/550 gives n = k' . (kQ)~lK*~*\ where k'
is a known constant coefficient for the chosen altitude. Then row
4 of the table gives V = JnD, row 5 the full b.h.p. available at
each speed, and row 6 the corresponding thrust horsepower (T.H.P.).
For the last we note, if the airscrew efficiency yj is not supplied, that
Y] = (//27c) . kT/kQ. To conclude this part of the table, row 7 may
record for future use the values of £pF*.

It is now necessary to arrive at Z>P, and the first step (row 8) is
to calculate Di = kff . X2/(?r . £pFa), where X = W/2s for straight level
flight and k" is a coefficient, probably in the neighbourhood of 1-1,
which is estimated from Chapters VII and VIII in consideration of
the plan-form of the wings and what allowance should be made for
wing-tip vortices. If test figures are available for the wings, row
9 will be devoted to the CL's, row 10 to the lift/drag ratios of the
wings, row 11 to their total glider drags (W -~- L/DW), and row 12 to
their profile drags DQ = Z)w — Dit Alternatively, row 8 may give
Cp,, row 10 CD, row 11 CDO and row 12 Z)0 = CDO . |pF2S.

If test figures are not quoted for the wings, we must estimate the

472 AERODYNAMICS [Cfl.

profile drags by direct calculation. For this purpose it will often
suffice to employ a constant profile drag coefficient obtained for
some Reynolds number intermediate between those for climb and
maximum speed. A first estimate results from calculating the
coefficient of flat plate skin friction by Article 220, after deciding
upon the probable transition Reynolds number from the wing
profile and surface in order to fix the second coefficient in (315), and
then adding a judicious increment (of the order of 25 per cent.) for
the thickness effect on skin friction with a like increment for the form
drag of normal wings. Whether wind-tunnel results or calculations
are resorted to, the question of surface roughness must be faced, and
some experience is necessary to forecast its effects (Article 229).

Row 13 records the part DP1 of the total glider parasitic drag
which is not affected by the slipstream, and row 14 the remaining
part Dpa which is to be increased on this account.

An isolated enquiry calls at this stage for choice to be exercised
as to further procedure. If the enquiry relates to cruising, range,
endurance, or the like, the actual slipstream drag in level flight
with the engine throttled back must be obtained. But interest
in preliminary calculations centres more frequently, perhaps, in
top speed and climb, for which the engine will be at full throttle.
Further description assumes the latter alternative.

Row 15 accordingly tabulates the maximum thrust T = 550 x
T.H.P./F for each speed. The factor by which £>P2 must be increased
is, for tractor airscrews, t = I + 4(a + a2), where a is an average
inflow factor appropriate to T uniformly distributed over an
effective airscrew disc area Se. 4(a + a8) = r/£pF2S,, the values
of which may be entered in row 16. Row 17 then gives DP2' = tDP2,
row 18 gives Dc = Di + DPl + DP2', and row 19 gives H' = DCF/550.
H' is not the power required for straight level flight except at top
speed ; at lower speeds it is the thrust H.P. necessary to overcome
the drag Dc when climbing.

The top speed is determined accurately by the intersection of
curves of maximum T.H.P. available and H' plotted against V.
The angle of climb 6 is given approximately by sin 6 = (T — DC)/W ;
the approximation is involved in row 8 of the table, since accurately
D% oc L* oc cos2 6. The error resulting from neglecting the difference
between cos1 6 and unity will usually be small but entails correction
for large angles of climb.

If H is required for cruising conditions, the only change is that
a should be calculated for a thrust T = D = Di + DP1 + Z)P2
[1 + 4(a + «')], which may be solved by successive approximation.

XI] PERFORMANCE AND EFFICIENCY 473

On changing the altitude, we notice that for the same values of
ipF*, i.e. the same indicated air speeds Vit D (but not Dc) will
remain constant if further scale effects be neglected. A unique curve
D against V% can therefore be plotted and values of D read off to
suit new calculations of V from / and kQ, which take account of
the supercharging of the engine and change of airscrew pitch.

The procedure can evidently be varied, and that described will
not always be the shortest. But it is typical and generally useful,
and has the advantage of tabulating familiar quantities, so that
arithmetical errors are easily detected.

Jet propulsion involves so great a change that methods described
in a later section of this chapter are preferable. To some extent
this is also true of airscrews driven by gas turbines and jet- airscrew
combinations.

261. Take-off and Landing Runs *
With the notation :

W = weight of aeroplane, L its lift, and D its drag ;
T = airscrew thrust and T0 that at top speed ;
V = velocity at any instant and F0 top speed ;
x = distance along the ground ;
IJL = coefficient of friction with ground ;

we have, during the run prior to take-off —
W dV

L). . (i)

Write as an approximation, a and b being constants

Now assume as a sufficient approximation that the tail is up during
the whole run to give top-speed incidence (error here would make our
estimates too small, but we shall find that they are too great), and
(i) becomes :

ldF_FW_2
g dt ~~ g dx ~~V
Introducing the constants A and B defined as —

A = a(T0/W) - \L

B = (1 + b) (T.IW) - VL.

* For flying boats see Gouge, Flight, November 1927, or Liptrot, Handbook of
Aeronautics.

474 AERODYNAMICS [CH.

(ii) simplifies to —

Integrating first to find t', the time from a standing start to take-
off at speed V —

,
' ' (373)

. A

the integral being a standard form.

Integrating alternatively from x = 0, when V = 0 to oc = x' at
take-off velocity F, to find the length of run —

' ' ' <374>

the logs, being of course to base e.

The following values for coefficients are in general use : a = 1-5,
b = 0-8, [i — 0-05 for average aerodrome surfaces and 0-03 for very
smooth, hard surfaces, such as the deck of an aircraft carrier (a
greater value is allowed over bumpy ground, for the energy then
dissipated in the shock absorbers is supplied from the engine) .

These simple formulae are very difficult to apply reliably. The
take-off speed would appear to be the speed for maximum climb, and
the initial rate of climb to be calculable by the methods of the pre-
ceding articles, due allowance being made for wind gradient (Article
85). But lift coefficients * and lift-drag ratios are so much larger at,
say, 15° incidence near the ground than in free air, that an aircraft
may be well clear before reaching its normal stalling speed. Tests f
of a number of aeroplanes have shown taking-off airspeeds to be less
than three-quarters that for maximum climb ; rates of climb at
30 ft. up have been recorded that exceed normal rates by one-third,
though a considerable part of this increase is due to wind gradient
and other retardation. Another factor of remarkable importance is
the distance of the C.G. of the craft behind its front wheels ; adding
to this distance by only 2 per cent, of the wing chord may increase
take-off run by 25 per cent. This is probably due to excessive
incidence, a similar disadvantage resulting from keeping the tail low
over rough ground to avoid risk of over-turning.

Unfortunately, different aircraft evince these non-calculable
effects in different measure, and to this uncertainty must be added
variation in pilots' skill. The specification of an aeroplane includes

* Mathematical investigations relating to this important problem have been carried
out by Tomotika, Nagamiya, and Takenouti, Tokyo Repts. 97, 1933 and 120, 1935.
t Rolinson, loc. cit., p. 122.

XI] PERFORMANCE AND EFFICIENCY 475

that it shall clear an obstacle height at a given distance from rest.
Where there is little margin to spare, designing for this requirement
calls for experience.

Many of the above remarks apply to landing runs. Force co-
efficients are high when undercarriages permit ; retardation is
greater on slightly rough ground, leading to less run, provided the
wheels hold the surface. In a difficult high-speed case the angular
and vertical motions of the craft after first touching must be worked
out from instant to instant in connection with the proper design of
the shock absorbers, smooth ground being assumed but variations
of Aerodynamic force being taken into account. All high-speed
craft require wheel brakes, which halve the landing run necessary
when a wheel replaces a skid at the tail. Their maximum retarda-
tion is usually of the order of 4 ft. per sec.8 With a skid and
no brakes the over- all coefficient of friction is about 0*1.

262. Range* and Endurance

The practical problem of how much fuel and oil an aircraft of the
aeroplane class must carry for a given length of non-stop flight is
beset with uncertain factors. Consequently, approximate analysis
is usually sufficient, it being understood that important factors will
be introduced subsequently.

We first investigate minimum fuel for a given range. This seems
to imply least work done, or drag a minimum, and with normal craft
about half speed ; but the engines will be throttled well back, and on
this C, the specific fuel consumption, depends acutely, while the
efficiency vj of the airscrews is also concerned, C is defined as the
Ib. of fuel per b.h.p. (H) per hour, and is assumed for present pur-
poses to include lubricant. The rate of variation of the gross weight
W in regard to distance % flown through the air is dW/dx = — CH/V.
We reckon (in this article only) V in m.p.h. so that x will be in miles,
and then v)H = WV 4- 376(1/0), LjD being the lift-drag ratio for
the complete craft. Thus —

Integrating on the assumption that the coefficient of W is kept
constant, and changing the log. from base e —

*0= 863-5 logloF . . (375)

* An interesting discussion of extreme range has been given by Fairey (Sir
Richard), Jour. Roy. Aero. Soc., March 1930.

476 AERODYNAMICS [CH.

where Wi is the initial gross weight and w the weight of fuel required
for a flight of x* miles, w is a minimum when 7](L/D)/C is a maxi-
mum. Unfortunately, this quantity cannot be regarded as constant
as assumed, although it can be evaluated at the beginning and end
of the flight, and the mean used. However, it must be understood
that under practical conditions of operation (376) is far too optimistic,
and that w requires to be increased very considerably for safety,
quite apart from risk of head winds, which is supposed to be taken
into account in #0.

To calculate roughly the minimum fuel required for a time <0
(hours) in the air, we may proceed as follows. The equation
dWjdt ~*~CH leads to :

CW 60

376Y}(L/D) ' 88

where S is the wing area, the factor 60/88 taking account of V being
in m.pJh. Hence :

*- 550

<# = — 550

Integrating on a similar assumption to that used above

giving for maximum endurance the condition y(LID)<\/^CLjC a
maximum. Again we note that average values must be taken.
When this is done and allowance made for operational difficulties, *0

^.yj

to ^ jA

PERCENTAGE DECREAS
OF FUEL CONSUMPTION (I
o S S S ^

0 5 10 15 20 25

ALTITUDE (THOUSANDS OF FT.)

z
t*j °4.n

»£10

UJ S

«??in

PERCENTAGE !N(
OF SPECIFIC CONS
=> S o jj

S"~

FIG.

10 0-8 0-6 0*4 0-2
THROTTLED B.H.R/NQRMAL

184. — THE AVERAGE ALTITUDE VARIATION is FOR NORMALLY ASPIRATED
ENGINES AT CONSTANT R.P.M.

XI] PERFORMANCE AND EFFICIENCY 477

may amount to little more than one-half what it would be if optimum
values could be maintained.

Altitude comes into both the above questions through C in the
first and \/P/C in the second. To include this, H .f(h) must be
substituted for H. Introducing x as an altitude factor for the
specific consumption, it is required that f(h)/xC be a maximum.
Fig. 184 gives some average values for normally aspirated engines.

AERODYNAMIC EFFICIENCY

262A. The efficiency now to be discussed differs in nature from
that described in Article 262, being the aeronautical adaptation of
a universal basis on which the economy of transportation may be
assessed. The load carried is compared with the work done, having
due regard to the speed achieved. For general purposes, e.g. com-
parison between aerial and surface transport, only the useful or
disposable load would be considered and the speed would be reckoned
relative to the ground. But the term Aerodynamic implies that
gross weight or total lift is substituted for useful or disposable load,
and true air speed for ground speed. These steps can be justified
from a general point of view, provided aircraft of abnormal tare
weights and exceptionally low speeds are excluded (cf. Article 69),
and they separate Aerodynamical from structural and mechanical
considerations.

Let W be the total weight of an aircraft, including its load, and
% the distance traversed horizontally through a still atmosphere at
a true air speed V. Regarding W and % as of equal value, the useful
result achieved is expressed by the product Wx. The work done
against the drag D is Dxt but Aerodynamical losses are incurred in
providing a thrust T = D. If the propulsive efficiency YJ takes
separate account of these additional losses, the total work done is
Z)#/Y], and the Aerodynamic efficiency is proportional to -yjJF/Z), i.e.
to Y)L/Z), since for straight and level flight W = L the lift.

Now clearly in the case of propulsion by engines and airscrews,

L - WV - 6 W (tons) x V (m.p.h.)

71 ~ ~~

D ~ TT/ij ~~ b.h.p.

very closely, and with jet or other propulsion ' b.h.p/ may stand
for the power of the propelling device apart from specifically Aero-
dynamical application. Hence, defining the Aerodynamic efficiency
of an aircraft by —

1 L ...

n*»ii. • • • - W

478 AERODYNAMICS [CH.

y)A is seen to be closely equal to ' ton-miles per b.h.p.-hour ' and
proportional to 'ton-miles per gallon/ In early days of heavier-
than-air flying, TQA provided a target of 100 per cent, which was
difficult for aeroplanes to surpass. But this position has long ceased
to hold, and V)A has become a figure of merit, having values usually
between 1 and 4.

In the first instance, Article 69 will be further developed in terms
of efficiency and with the detail now possible.

It follows immediately from that article that for geometrically
similar airships of size / and using the same gas, L/D oc Z,1/3/Ft2.
Thus plotting TJA/Y) against V? gives an hyperbola for each size.
This basis is adopted in Fig. 184A in order to penalise low speeds
and give weight to high speeds. The family of curves for various
sizes has been prepared by scaling down known data for large
airships.

It has also been seen that, considering a series of geometrically
similar aeroplanes of span I in straight and level flight, their drag
through the major part of the speed range can be expressed approxi-
mately as —

and is a minimum when each of the terms on the right is equal to
L \^AB, which occurs at an indicated air speed given approximately

'- "£

Thus the maximum L/D is l/2^/ABf and at any other speed —
L ___ 1 __ 2(L/D) max.

For cantilever monoplanes having * normal * wings of aspect ratio
8, a maximum L/D of at least 18 can be expected with airscrews
feathered to give zero thrust, and then adding 8 per cent, to the
theoretical minimum for the induced drag leads to the following :

(AB)W= 1/36; A = 2-16/w ; F,0 = 24-6^1/2 (m.p.h.) ;

F0* = 1300 w/a (ft. per sec.)2 ;

where w is the wing-loading in Ib. per sq. ft. and cr the relative
density of the air. These data are used for illustration below.
The straight line (a) of Fig. 184A represents the constant

XI] PERFORMANCE AND EFFICIENCY 479

maximum efficiency for aeroplanes of the given shape loaded
between 20 and 50 Ib. per sq. ft. The optimum true air speeds are
unduly small at low altitudes even with heavy wing-loadings and,
as previously noted, aeroplanes fly faster, losing efficiency but
putting to use engine power provided for climbing. The expression
(ii) and the above numbers lead to the curves given for w = 20, 30,
40, and 50 Ib. per sq. ft., showing the drop in efficiency at higher
speeds with these wing-loadings. The advantage achieved by
decreasing the size of an aeroplane for a given total weight may be
compared with that secured by increasing the size of an airship.
Improvement of aeroplanes by this means is restricted by take-off
and forced landing conditions (minimum flying speeds for full load
vary from 56 to 88 m. p. h. through the above range of wing-loading,
assuming a maximum CL of 2J). Improvement of airships by
natural saving of surface area with increase of size is seen to become
slow at the 200-ton stage. Other conclusions arrived at in Article
69 may be verified from the figure. The curves so far described
apply to all altitudes if abscissae are taken as indicated air speeds.

2628. It is of interest to trace the effect of wing-loading on the
efficiency, in straight and level flight at full power, of aeroplanes
having specified initial rates of climb at F0. With airscrew effects
excluded, these rates will differ little from maximum rates of climb.
Let v be the rate in feet per minute. Then approximately —

W f V_ __ VQ 1 Wv

550 LZ/D ~~ (LfD) max.J ~~ 33000*
if V is the top speed, or by (ii) of the preceding article —

Writing k for V0*/w results in the following equation for V —

which is in suitable form for solution by successive approximation.
The above are restricted to true air speeds and cannot be interpreted
as indicated air speeds.

Considering the example of Fig. 184A with cr == 1, the efficiency
curves (b)-(e) are obtained for full speeds at low altitudes corre-
sponding to initial rates of climb of 500, 1000, 1500, and 2000 ft.
per min. at values of F0 appropriate to the wing-loading. These
curves are not operational but indicate the advantage of designing
for high wing-loadings. Thus doubling the wing-loading from 20

480

AERODYNAMICS

[CH.

50 100 150 200 250 300

AIR SPEED 1NM.PH. (INDICATED EXCEPT FOR CURVES (bj-ie)j

FIG. 184 A. — AERODYNAMIC EFFICIENCY OF AIRSHIPS AND AEROPLANES.
v » Initial rate of climb, w = Wing loading. W = Gross weight of airship.

to 40 lb. per sq. ft. increases the ton-miles per gallon by about
16 per cent, for the same initial rate of climb, in spite of the top
speed being increased by some 30 per cent.

262C. Airscrew Effects

A close network of curves of Y)A/Y) against Fa for constant wing-
loadings and initial rates of climb provides a chart which may be
used to criticise a given design or aid in a proposed one. The few
curves for constant wing-loading in Fig. 184A include allowances
for form drag and wing-tip vortices in the values assumed for the
maximum glider L/D and A. They follow directly from these
values by ignoring Aerodynamical scale effects, but may equally be
plotted from an experimental curve of L/D against CL for the given
geometrical shape. Allowances have not been made, however, for
slipstream effects or airscrew efficiencies, which reduce Aerodynamic
efficiencies and rates of climb.

Thus taking, for example, the average lightly loaded aeroplane
of merit described in Article 252, the known speed of 220 m.p.h.
and wing-loading of 21 lb. per sq. ft. indicate on the chart a value
of YJA/Y] of about 1-48. From the known power-loading (12-3 lb.
per b.h.p.) — TQA __ ton-miles per hour

== ~" ~ r ' '-^ J."4fe/j

TJ 73 x b.h.p.

XI] PERFORMANCE AND EFFICIENCY 481

on giving YJ the reasonable value 0*82. So far as may be deter-
mined, this typical meritorious aeroplane satisfies the chart in
respect of efficiency at top speed in spite of slipstream drag. But
the idealised rate of climb is seen to be more than 1600 ft. per min.,
whilst the actual rate would be 300-400 ft. per min. less. The
large deficit will now be investigated.

Slipstream Drag. — We first determine the decreased lift /drag ratio,
denoted by LJDV for straight and level flight at the general speed
V. Neglecting change of induced drag, a fraction / of the total
parasitic drag is increased by the factor 1 + 4 (a + a2), where a is
the inflow factor determined from the thrust T on an effective air-
screw disc area Se. From airscrew theory, 4(a + #2) = 2T/pV*Sll.
Hence the total parasitic drag is increased by the factor 1 + 2fT/
pF'S, and, by Article 262 A,

L __ 2(LfD) max.

Z>! ^ /F0y ~~7~7y 2} w
\v) + \yj + plv

Lf fW

where L/D is the glider lift/drag ratio given by (ii) of Article 2 62 A.

For a given aeroplane and load, the percentage reduction of the
glider lift/drag is the same for all speeds. For a constant geo-
metrical shape, pF0aoc w and the second term in the brackets of
(i) oc fS/Set where 5 is the wing area. The variation is obviously
discontinuous and complicated. For the single-engined aeroplane
with W = 10,000 Ib. and w = 20 Ib. per sq. ft., / might be equal
to i and S, to 80 sq. ft., giving LjDt = 0-946(Z,/Z>). Doubling
the weight without changing the size would have no effect unless
a second engine were added, when /might become £ and Sf 160 sq. ft.,
giving L/D! = 0-972(L/Z)). Halving the wing area with the
original weight and a single engine might increase / to f but reduce
fS/Sg by 25 per cent. Such examples verify that the effect on
efficiency of slipstream variations is comparatively small in straight
level flight, but it is worth remarking again that the absence of
tractor airscrews altogether might improve efficiency considerably
by increasing transition Reynolds numbers ; such improvement
would appear in the present calculations as a greater maximum
lift/drag ratio.

Turning to climb at V0, the large thrust is approximately equal
to £>a + Wv/QOVQt where Z)a = W7(L/Z)a) and is obtained from the

A.D.— 16

482 AERODYNAMICS [CH.

glider drag D at that speed by increasing the total parasitic drag by
the factor —

, J!_\

"*" 60 F0/-

Thus, writing F for the factor outside the brackets, we readily find
from Article 2 62 A that —

2LD max-

60 F0

For a 10,000-lb. aeroplane of the series to which Fig. 184A
applies, and with w = 20 Ib. per sq. ft. and/ = £, F = 2, closely.
Putting v = 1500 ft. per min. gives L/D2 = 14-7. The maximum
lift/drag in level flight is reduced by slipstream drag to 0-944 X
18 = 17, whence it is easily found that the increase of slipstream
drag from top speed to F0 reduces the rate of climb in this case by
about 90 ft. per min., whilst the entire slipstream drag accounts for
a loss of about 120 ft. per min. The remaining part of the decrease
in rate of climb from the ideal value is to be traced to the loss of
thrust h.p. by airscrews and is minimised by use of variable pitch,
as already described.

2620. Application to Prediction

A curve of Y)A/TQ = ^LjDl against the indicated air speed Vi
(= a1/2F m.p.h.) for a given aeroplane in straight and level flight is
plotted by reducing the corresponding glider curve by a factor to
allow for slipstream drag. Such a corrected curve for an aeroplane
is shown in Fig. 184B and, by (ii) of Article 262 A and (i) of Article
262C, it is independent of altitude.

Let P be the actual power-loading in Ib. per operative b.h.p.
By Article 262A the conditions L == W, T = Dl for straight level
flight are satisfied if —

w (tons> x v*

Yj\/a X b.h.p.
PVi

Let any point A on the efficiency curve subtend at the origin the
angle 0 with the Fraxis. Then the value of C required by the
wing-loading and speed specified by A is equal to (tan 6) xfy, the

XI]

PERFORMANCE AND EFFICIENCY

483

FIG. 184s.

scale of 7)A/Y) being % units of length and that of Vt y units of length.
Thus the indicated air speed for known values of P, 7) and a can be
found by drawing a radial line from the origin at the angle 0 ==
tan"1 Cy/x to intersect the efficiency curve. More generally, in-
formation regarding the power units enables CV4 to be plotted
against Vi in Fig. 184B as a sequence of curves, one for each
altitude, and the intersections A, Av A& . . . give the efficiencies
and speeds of flight for those altitudes.

Increase of P (reduction of b.h.p.) and decrease of <r both
increase 6, and the absolute ceiling occurs, when the radial line is
tangential to the efficiency curve, as illustrated. Writing k4 for
vVffjw and V i for the indicated air speed in feet per second gives the
following equation in place of (iii) of Article 262B —

whence the rate of climb at ViQ can be calculated for any altitude
at which the top speed in level flight is known. A correction for
the slipstream can be effected as already discussed.

484

AERODYNAMICS

[CH.

262E. Wing-loading and High-altitude Flying

Referring again to Fig. 184B, if P and YJ can be regarded as
constant up to a certain altitude, C oc I/ \Ai through the range, and
a convenient construction gives the true air speed. Let the point A
in the figure represent low-altitude flight, and use suffix 1 to distin-
guish flight at a higher altitude. Then if Vl is the true air speed in
m.p.h. —

/~ l~\ 17 4^o^ ft

L = tan 6.

Hence Bv the point on OA produced which has the same value of
W7) as ^i» giyes the true air speed Vl on the scale of F,.

The assumption will now be extended to a large increase of
altitude and a considerable range of speed. In this form it will be
representative of the reciprocating engine and airscrew, even
approximately, only up to the supercharged height, but super-
charging to a high altitude will also be assumed. Turbine, jet,
rocket, or composite power units may make the assumption more
widely representative.

In Fig. 184c, the pencil of lines radiating from the origin, marked

KK)

200 300

MILES PER HOUR

400

500

FIG. 184c.

XI] PERFORMANCE AND EFFICIENCY 486

sea-level to 40,000 ft., is appropriate to a power-loading of 10 Ib.
per b.h.p. Intersections with the ideal efficiency curves given
for w = 20 and 50 Ib. per sq. ft. lead, by the construction just
described, to the true air speeds inset as full-line curves at the
bottom of the figure. Thus the speeds marked on the scale are to
be interpreted, in respect of the upper and left-hand part of the
figure, as indicated air speeds, and as true air speeds in respect of
the lower and right-hand part. Corrections have not been made for
slipstream drag, but previous discussion has shown that they would
affect the results only in degree.

The main feature evinced is the large increase of efficiency
achieved at high altitudes with, at the same time, a large increase
of true air speed. To the present approximation, a wing-loading of
20 Ib. per sq. ft. gives at 28,000 ft. altitude the same efficiency and
speed as a wing-loading of 50 Ib. per sq. ft. at sea-level. At 40,000
ft. the relative density of the air is |, and this altitude is often
regarded as suitable for long-distance flying with pressure cabins.
It permits an aeroplane with a wing-loading of 50 Ib. per sq. ft. to
fly within 15 per cent, of the maximum possible efficiency and at
twice the true air speed for the same efficiency at low altitude.

The dotted radial beyond the pencil of related lines applies to
40,000 ft. with the assumption that power is provided by reciprocat-
ing engines supercharged to rather below 30,000 ft. Associated
changes in the altitude-speed curves between 30,000 and 40,000 ft.
are shown dotted. The loss of power brings the aeroplane with the
heavier wing-loading close to its ceiling, and the serious loss of
speed ensuing at 40,000 ft. illustrates the advantage of jet or other
propulsions in which the difficulty of maintaining power at high
altitudes largely disappears.

262F. Laminar Flow Effect

As an example of a general kind we may consider briefly the
improvement of the pre-war type of monoplane used above for
illustration. Technical accuracy will not be attempted, the aim
being to assess by simple means only the order of gains in efficiency
up to the laminar flow stage. A Reynolds number of 15 million will
be assumed for calculations and scale effects through the upper speed
range neglected. Skin friction will be estimated by (315), with an
allowance of 25 per cent, for thickness and a like addition for form
drag in the case of normal wings, as suggested in Article 200A. The

486

AERODYNAMICS

[CH.

following approximate formulae, which can easily be verified, are
adopted from Elementary Aerodynamics —

•35'

F,0 = 16-5

(L/D) max. w
~~~A

(ii)

A denoting the aspect ratio (taken as 8) and CDP the coefficient of
total parasitic drag. The method being suitable for no more than
a first approximation, only an outline of the calculations will be
given and round numbers used wherever possible.

(1) For the aeroplane in its original state, (L/D)max. = 18 and
(i) gives CDP= 0-0176. The transition Reynolds number is low
on account of roughness and slipstreams, and the assumption of
1 million leads to the estimate Cpo^ 0-008, without increase for
roughness drag. This leaves the value 0-0096 for CDB, the coefficient
of extra-to-wing drag, including roughness wherever occurring.
Contributions to CDB are assumed to be distributed as indicated in
Table X, p. 459.

(2) Let the first improvement consist of eliminating roughness,
tractor airscrews, non-ducted reciprocating engine cooling, and an
exposed tail wheel. Inspection of Table X suggests a reduction of
CDB by about 25 per cent., i.e. to 0-0072. Another consequence is
to increase the transition Reynolds number for the ' normal ' wings
to, say, 3 million. The revised estimate of CDO is 0-0064. Hence
CDP becomes 0-0136 and, by (i), (L/D)max. = 20-5.

(3) Let the second improve-
ment be the substitution of
laminar flow wings with transition
at f chord behind the nose.
Whether the form drag is greatly
less on increasing the transition
Reynolds number from 3 to 10
million cannot be decided without
special data, and the assumption
above will still be made. Then
CDO = 0-0032 and CDP = 0-0104,
whence (i) gives (L/D)max. =
23-4.

Assuming further a wing-loading of 36 Ib. per sq. ft., (ii) gives
approximately for the most efficient speeds : Fi0 = 148, 158 and
169 m.p.h., for (1), (2) and (3), neglecting scale effects.

300

Vv mph

FIG. 184D,

XI] PERFORMANCE AND EFFICIENCY 487

Efficiency curves for the three cases can be constructed from these
data very rapidly, as described. In Fig. 184D the percentage
increase of TQA/TQ for the two improvements is plotted against the
indicated air speed. Normal wings are retained for the dotted
curve, and two alternative locations are shown for the local increase of
efficiency due to laminar flow wings, the one to the left in the figure
being suitable for high altitude flying. The extent of the alternative
favourable ranges is conjectural and remains matter for design.
The figure illustrates that laminar flow wings are the more effective
at small lift coefficients, as is otherwise evident.

To derive corresponding curves of Y)A, we should require to take
account of the variation of propulsive efficiency, not only in regard
to change of speed and altitude but also as involved in the sweeping
alteration introduced between cases (1) and (2).

263. The Autogyro and Helicopter

Fig. 185 is a sketch of a recent autogyro (Articles 243-4). An
earlier type of these small craft has, additionally, diminutive wings
with ailerons totalling some 5 per cent, of the rotor disc area. One
example of this kind, of 0-7 ton all-up weight with a disc area of

FIG. 185. — WINGLESS DIRECT-CONTROL AUTOGYRO.

900 sq. ft., gave approximately: top speed 97 m.p.h., with 110
b.h.p. ; minimum horizontal speed 40 m.p.h. ; minimum vertically
downward component of speed 10 m.p.h. at a gliding angle of 16° ;
normal rotor speed 180 r.p.m. Further development is still in
progress, and these figures cannot be taken as indicating either best

488 AERODYNAMICS [CH.

modern results or ultimate scope. Rotor solidity used to be as high
as 0- 14, then 0-07, and is still decreasing. Success with direct control
tends to eliminate the auxiliary wing, though the combination has
points of Aerodynamic interest. Preliminary prediction of autogyro
performance follows lines already established for the aeroplane, once
rotor characteristics are available from the theory of Chapter X or
from model experiment freed from scale effect. It is only necessary
to discuss in general terms the main differences to be expected.

The stalling disc incidence of a rotor is large, about 40°, but its
lift/drag ratio, Ct/Cxt is then very small, about 1-2, and minimum hori-
zontal speed is determined by engine power. Though this condition
occasionally applies to aeroplanes (Article 77), their wings are then
about to stall, while the autogyro has a large reserve.

We can verify this roughly for the above example by neglecting all
parasite drag and auxiliary lift. C, is found to be 0-42 at 40 m.p.h.,
corresponding to 13° incidence ; Ct/Cx would then be about 3-7,
giving drag = 420 lb., which would require an effective thrust h.p.
of 45. That available would be about 50, leaving little margin.
But a maximum C, approaching 1-0 could be expected, so that
stalling speed = 26 m.p.h. The craft could realise such low speeds
in descent, gravity supplying the power required. One condition
could be a vertical path, as the stall is not catastrophic. But
minimum speed of descent, worked out as described in Chapter IV,
would correspond to a higher forward speed and a moderately flat
gliding angle, as mentioned above.

It is seen that the autogyro can use increase of power to increase its
speed range by decreasing minimum sgeed. Also, with a quite
feasible undercarriage, it can land safely in an extremely confined
space. Together with use of impulsive helicopter lift as already
described, to give direct take-off, these features form the distinguish-
ing advantages of the type.

Turning to rate of climb, the type is greatly inferior to the aero-
plane class, owing to very poor efficiency at the forward speed giving
maximum reserve power for climbing. The L/D of a complete auto-
gyro is then little more than 5, and consequently even an inefficient
aeroplane of the same weight and power climbs twice as fast.
Maximum climb will be found to occur at about two-thirds speed, as
with aeroplanes.

At greater speeds the L/D of the rotor rises, attaining a maximum
(say 8) just before top speed is reached. At present, the resulting
decrease of rotor drag approximately offsets increase of parasite
drag, so that over-all L/D is ultimately little less than at maximum

XI] PERFORMANCE AND EFFICIENCY 489

climb. This essential difference from the aeroplane, whose efficiency
declines from maximum climb onward, enables the autogyro to bear
comparison somewhat better at full speed, but the low LjD of about
5 still remains as an important disadvantage. Top speed L/D can
doubtless be increased, but aeroplanes also have greater efficiency
in prospect. Eventually, the penalty of deriving sustentation from a
screw motion of the lifting surface must remain fundamental.

These comparisons between the autogyro and aeroplane are in-
evitable, but no more justifiable than between the airship and
aeroplane. Multiple power-unit aeroplanes unquestionably pro-
vide the most efficient and the safest means of high-speed transport
that at present it is possible to conceive. In this duty we have
already seen that the airship cannot compete. A similar disability
arises with the autogyro from the fact that disc loading must increase
rapidly with increase of speed, leading to less efficiency and high
minimum vertical velocity. The same need to increase loading
exists with aeroplanes, but not with the same consequences. The
high-speed aeroplane demands large, prepared aerodromes, a
peculiar disadvantage which the autogyro and airship both avoid,
the former so successfully as alone to justify this remarkable inven-
tion. Additionally, the autogyro, owing chiefly to non-stalling
properties, is easy to fly in a straightforward way, although safety
in manoeuvres remains a matter for investigation.

A sketch of a helicopter is included in Fig. 43. It is of interest
that this type of aircraft was the first to be invented (Leonardo da
Vinci, circa A.D. 1500) and has been the last to receive practical
form, a step partly due to the success of the autogyro. The example
illustrated has only one lifting rotor, but a subsidiary rotor, work-
ing in a vertical plane at the tail, controls orientation of the body.
In another type this control is obtained from a small jet utilising
the exhaust gases of the engine. Theoretically, the helicopter is
slightly more efficient than the autogyro, but until the type is
further developed and performance data become available, it can-
not be concluded whether this small advantage will be realised.

CORRECTION OF FLIGHT OBSERVATIONS

263A. The performance of a given aeroplane depends acutely
upon the state of the atmosphere at the time of the test. In order
to assess the capabilities of the aircraft, observations have therefore
to be reduced to a common basis, and some standard atmosphere is
chosen, such as that defined by Table III, p. 17. Methods of reduc-
A.D.— le*

490 AERODYNAMICS [CH.

tion are easily devised, and the following indicates one procedure that
will often be found suitable.

Maximum Speed in Level Flight. — Data from speed tests in straight
and level flight, whether automatically recorded or observed by the
pilot, can be tabulated under the following headings :

TABLE A

Aneroid
height.
hA (ft.)

Atmospheric Indicated
temperature. air speed
6° C. Vt (m.p.h.)

Airscrew
revolutions.
JV (r.p.m.)

The first column of the table consists of a number of selected
altitudes as given by the altimeter, and the other columns record
averaged data for flight at each of these reputed altitudes. The last
column will be omitted in case of jet propulsion, but airscrews will
be assumed for greater generality.

The altimeter is a pressure gauge, and the pressures are correctly
written down from Table III, identifying hA with the true altitude
for that purpose. But the corresponding densities would be in-
correct unless the second column of Table A happened to accord with
the standard atmosphere. The true values of the relative density
are obtained from

288 p

Po 273 + 0 p0'

in which the suffix refers to conditions at the foot of the standard
atmosphere, where 6 = 15° C.

It is necessary to determine the brake horsepowers actually
expended at the various aneroid heights. These can always be
found from records of ' bench ' tests carried out on the power units
at various air pressures, knowing the pressure, density, and tempera-
ture during flight. For instance, reciprocating engines often give,
for normal aspiration or above the rated altitude, a linear relationship
between the b.h.p. and the variable ^/(^p/pQ) at constant r.p.m.,
and the actual b.h.p. can be read by interpolation from charts
prepared on this basis.

On multiplying the b.h.p. and N by y'a in accordance with
Article 259, a new table with headings as follows results from
Table A :

TABLE B
PlPo ff Vt N\/G b.h.p. -y/a.

XlJ PERFORMANCE AND EFFICIENCY 491

The original data have now been expressed in terms of the actual
pressure, density, and temperature during flight, with the actual
horsepowers added.

Let any true altitude h be chosen. Corresponding values of
pt p, and 6 appropriate to the standard atmosphere are written
down from Table III, and the b.h.p. available at any r.p.m. imme-
diately follows. The values of b.h.p. \/a and N\/G required to
satisfy Table B with the new values of a and 0 are then found by
plotting, and the solution gives the corresponding indicated air
speeds without further work. The calculations are repeated for
other true altitudes.

Maximum Climb. — Flight records regarding climb are presented
in the foim :

TABLE C

hA 0 V, N Time to hA

t (min.)

The aneroid rate of climb is readily found, with the help of (99)
p. 145, for heights above the rated altitude. It is required to deduce
true rates of climb at true altitudes. It is assumed in the following
that the rows of Table C differ by only short intervals of hA and t ;
otherwise the records may be plotted and small changes read from
the curves. The nature of the correction required may be visualised
from the reflection that an aneroid rate of climb would be registered
for level flight if the density decreased at the altitude concerned.
Through a restricted change of height, the pressure and density
are related by the hydrostatic equation (2), p. 5, i.e. AA = — A^>/pg.
The mean value of p during a short interval of time is readily deduced
from changes in the altimeter and thermometer readings, by use of
the equation of state (9), p. 13. Hence the actual climb AA follows,
and we have, if or' is the standard relative density used in calibrating
the altimeter,

po = A/*A<r',

whence the true rate of climb is given by

dh a' dhA
dt a dt

This equation may be used to correct the recorded time intervals
to standard atmospheric conditions, and then the true time to a
given altitude, or required to traverse a given change of altitude,

492

AERODYNAMICS

[CH. XI

follows by summation. The graphical representation of altitude
plotted against time is often called the climb diagram (cf. also
Article 81).

The remaining data of Table C can now be used to reduce the
corrected rates of climb to standard atmospheric conditions by
obtaining the horsepowers available at various true altitudes under
those conditions, the calculations being the same as for maximum
speed in level flight. A graphical solution for the new values of
a gives corresponding values of V,, N*\/a and the rate of climb
x\/<J, whence Table C may be standardised.

Plotting. — In Fig. 185A (a), curve 1 is plotted from the last two
columns of Table B, whilst curve 2 is obtained from the engine data
for any chosen altitude in the standard atmosphere by assuming
some likely values of N. The intersection determines the b.h.p.

(a)

Fig. 186A.

required for straight and level flight at maximum speed under
standard conditions. The speed is read from (6) in the figure,
which is also plotted from Table B. Repetition for other altitudes
standardises maximum speed performance. Simple development
extends the method to climb.

Chapter XII
SAFETY IN FLIGHT

264. Complete discussion of aerial safety would involve such
matters as engine reliability, structural design, life of materials, and
the direction of flying routes. Aerodynamics is concerned mainly
with three other factors, as will be described.

A craft should be capable of flying itself, in the sense of maintain-
ing a particular mode of flight, for which its controls have been
arranged, without further assistance from the pilot. This faculty
might be secured by a mechanical or robot pilot, when the craft is
said to be automatically stable. But we shall study only the case
where no substitution for the pilot is necessary, the craft possessing
an inherent stability by virtue of judicious shaping and distribution
of its mass. Inherent stability in aeroplanes must be limited by
both flight variation and violence of disturbance. Beyond this
range, and during manoeuvres, safety against disturbance lies in the
hands of the pilot, and depends upon the provision of controls which
will remain adequate in rather extreme circumstances. The third
Aerodynamic consideration is concerned with specifying what
accelerations are to be expected from the response of a craft to
disturbances (such as gusts) or the reasonable exercise of its controls,
so that the structure may be designed to have sufficient strength.

After introductory articles, we shall proceed, in the first place, to
rigorous study of inherent stability in straight flight. The theory
and application of the method are due, following pioneering work by
Lanchester and others, to Bryan,* and to Bairstow f and his
collaborators at the N.P.L. As shown by the dates given, the theory
is one of the oldest of Aerodynamics. But, though not difficult
mathematically, the subject is very complicated, and only recently
has it been recast into a form suitable for discussion and use by the
designer. This step is due to Glauert. J

Aeroplanes are examined theoretically for response to very small

* Stability in Aviation, 1911.

f Adv. Com. for Aeronautics, 1912-13 ; also Applied Aerodynamics, 1920.
| A.R.C.R. & M., 1093, 1927. For the most recent general account see B. M.
Jones, Aerodynamic Theory, vol. v, 1935.

493

494 AERODYNAMICS [CH.

disturbances, and practical utility depends on the assumption that
they will behave similarly in face of the disturbances encountered in
normally bumpy weather. This assumption is sanctioned by full-
scale experience ; it fails in some cases, but usually for specific
reasons that are apparent on inspection. Maintenance of flight is
not continuous, a stable craft requiring time to recover from dis-
turbance. Recovery is achieved by a natural manoeuvre, in which
the C.G. is displaced from the course of mean motion, variable linear
and angular velocities being superposed. Damping out the effects
of disturbance may occupy a fraction of a second or more than a
minute. Space is required and, when this is lacking, as on landing,
it is essential for a pilot to be able to supersede stability by control.
Even with no restriction on freedom of movement, a tedious response
by a stable aeroplane will usually be corrected by control at an early
stage. Thus, stability moments should be light and easily over-
ridden. But this requirement follows also from the fact that too
strong a response to one kind of disturbance may involve instability
in regard to another. Thus, designing for stability does not lend
itself to the employment of large margins to cover error, but calls for
careful compromise between conflicting factors. Evidence of static
stability is often a poor guide to the possession of the dynamic
stability with which we are now concerned.

Longitudinal stability deals with changes in the plane of sym-
metry, such as of pitch or air speed ; lateral stability includes all
asymmetric movements, such as roll or sideslip. Great simplifica-
tion follows mathematically from the assumption of initially straight
flight : longitudinal and lateral stability do not affect one another,
and may be discussed separately. We do not attempt to follow the
complicated motion that would develop in an unstable craft without
control : we are concerned only with the way in which instability,
if any, first occurs and from what causes. A large number of
factors affect stability, and all must be retained in the examination
of a border-line case. But several are of little importance, and may
be omitted from the approximate solutions that often suffice in
practice.

265. Axes and Notation

The motion of the C.G. of an aeroplane is determined by the
resultant force and the rotation by the resultant couple about the
C.G. We use a right-handed system of axes (Fig. 186), with the
origin at the C.G. : Oz lies in the plane of symmetry and is directed
approximately downwards in normal flight ; Ox points forward and

XII] SAFETY IN FLIGHT 495

FIG. 186.

Oy to starboard. These axes are fixed relative to the craft and
move with it. Ox may be chosen (whence the others follow) as a
principal axis of inertia, or arbitrarily (e.g. parallel to the wing chord
or the airscrew axis), or parallel to the undisturbed motion. The
last choice is usually most convenient, and the axes are then known
as wind axes. Positive pitch increases angle of incidence, a, positive
bank (<£) and yaw (<\>) are those for a right-hand turn, and positive
moments increase these angular displacements. Moments are
denoted by L about Ox (rolling), M about Oy (pitching), JV about
Oz (yawing), and the corresponding angular velocities by p, q, and
r . The components of Aerodynamic force in the directions of the
axes are Xt Y, and Z per unit mass of the craft. The transverse
moment of inertia, approximately about Ox, is denoted by A, the
longitudinal by B, and the directional by C.

V represents the resultant velocity of the aeroplane and U or «,
v, W or wt velocity components along the axes. ut v, w are always
small, but 8(7, 8w may be used for infinitesimal increments. W is
also used for the weight of the aeroplane, but no confusion will
arise. A small increase of incidence — w/V, a small positive angle
of yaw = — v/V.

Damping Factor. — Let wl be the initial value of a disturbance w,
which varies exponentially with time. Writing w = w^*, X (or its
real part) is called the damping factor. Assuming the motion to be
damped, let us calculate the time at which w will have decayed to
\i»i. Taking logarithms, log £ == M, or —

0-69

For the disturbance to be damped, the real part of X must be nega-
tive, and then the time to half-disturbance varies inversely as the
damping factor. If the motion is unstable, the time to double
disturbance follows in a similar way.

496 AERODYNAMICS [CH.

If X be wholly real, the disturbance decays or grows continuously,
and the motion is called a subsidence or a divergence, respectively.
If X be complex, let X = A + iB. Then

p = e(* + iB» = e"(cos Bt + i sin Bt)

and, as t increases, 4* is seen to oscillate in value with a period
2n/B and an amplitude e**. Thus if A — i.e. the real part of X —
be negative, the amplitude of successive oscillations decreases.
The disturbance is then damped, the time required to halve the
amplitude being — 0-69/^4. A positive value of A indicates increas-
ing amplitude and an unstable motion. Thus if X be complex,
its real part is the damping factor. It is possible for A to be zero,
when the oscillation maintains constant amplitude ; in accordance
with the preceding article, we regard this case also as indicating
instability in connection with flight.

Immediate Notation. — Numerous symbols will be defined later as
they arise, but the following are required for immediate use :
W = Weight of aeroplane.

S = Area of wings (of chord c).

I = Effective lever-arm of tail plane about C.G.

jx = W jgpSl, called the relative density of the aeroplane.
5' = Area of tail plane.

T = S'l/Sct called the tail volume ratio.

a = Slope of lift curve (dkjda) for wings.

a' = Slope of lift curve (dk'Jda?) for tail.

Oj = Tail setting angle (angle between wing and tail chords).

e = Angle of downwash.
Jcm = Pitching moment coefficient : M/pF*Sc.

k = Radius of gyration about transverse axis through C.G.

INTRODUCTION TO LONGITUDINAL STABILITY
266. The Longitudinal Dihedral

A necessary, though insufficient, condition for longitudinal
stability has been described in Article 87 : the rate of increase of
— M due to the tail must exceed that of M due to wings, body, etc.
Neglecting all forces other than the lifts of wings and tail, and con-
sidering only normal incidences for which a, a', and de/doL remain
constant, the argument may be arranged in convenient terms. It
concerns the upward vee, or longitudinal dihedral, between wings
and tail. Geometrically this = a,, but Aerodynarnically it is more
complicated.

SAFETY IN FLIGHT

497

XII]

If the C.G. is distant be behind the quarter-chord point (the
' Aerodynamic centre ') of the wings and km0 is the wing moment
coefficient at the incidence <x0 of zero lift, we can write for the wings
only —

For equilibrium at V and a —

Tail lift X -^-=- = AL,O 4- bk*.
pF*5c

On assuming a symmetrical section for the tail plane, so that its lift
vanishes at zero local incidence, this expression leads at once to
(cf. Article 87)—

ra' (a + a, — e) = km0 + ba(a — a0).

On transference to the L.H.S., km0 can be represented by an
additional angle e of the tail plane, giving —

T#' (a + a, — c + *) = ba(<x, — a0) . . (i)
Existence of a righting moment requires the L.H.S. of (i) to in-
crease more rapidly with a than the R.H.S., i.e. differentiating with
respect to a —

'(^ ds\ * t^

™\l~Tj>ba' • • ' (11)

or from (i) again —

Equilibrium-^

i.e. —

— a, > oc0 + e. . (iii)

In Fig. 187 the tail
moment curve is plotted
with its sign changed, so
that intersection with
the wing curve repre-
sents zero resultant
moment and equilibrium
on this account. The
apparent or geometric
dihedral— a, is increased
first by — QCO (<x0 being
negative) and then by e,
to give the effective
dihedral, for only when
the wings are at incidence a0 — e will their moment vanish.

e ' «0 0 <*t «

FIG. 187. — THE LONGITUDINAL AERODYNAMIC

498 AERODYNAMICS [CH.

It will be seen that (iii) is independent of downwash, assuming
de/dv, constant. However, from (ii) the righting moment is propor-
tional to —

and the factor within the brackets is about 0-65 for normal mono-
planes and 0-5 for biplanes (Articles 189, 192), so that S' must be
increased on this score. The same principle holds for ' tail-first '
aeroplanes, an old type recently revived and improved, but their
forward stabilisers work in an upwash which, in contrast with down-
wash, increases efficiency (usually by some 7 per cent.).*

It is to be noted that the simple idea developed in this article,
though useful in connection with certain compact types of craft,
cannot be applied directly to modern high-speed monoplanes, in
which the unstable moment of body and engine nacelles alone may
easily exceed that of the wings. f

267. The Short Oscillation

The foregoing considerations are static, the C.G. being constrained
and angular velocity zero. We now remove these restrictions, and
examine in a preliminary manner J the initial response of a stable
aeroplane in straight horizontal flight to a transient vertical gust.
The two moments of the last article, together with any other pitching
moments arising on the craft, are assumed added algebraically to
give a resultant static moment M, of which the coefficient is km. It

is clear that the craft will nose
into the relative wind. The C.G.
receives a vertical acceleration,
l 0 while there also occur changes both
of pitch and pitching.

Shortly after the impulse, when
Ut w are the velocity components
along Ox, Oz, let the craft pitch
through the small angle Soc in the
short time 82, the directions of the
axes changing from Ox, Oz to Ox',
Oz' (Fig. 188), while the velocity

;z'

components increase to U +

FIG. 188. and w + 8w. Resolving in the

* Bryant, Aircraft Engineering, July 1933.

t Cf. Lachmann, R. Ae. S.f 1936.

J The approximate treatment in this article is based on a paper by Munk.

XII] SAFETY IN FLIGHT 499

original direction Oz, the increase of velocity in 8* is Sw cos 8oc —
(U + 8C7) sin 8oc = $w — £78a to the first order of small quantities.
Hence the acceleration in the direction Oz comes to dw/dt — U . dafdt.
V and q can be written without serious error for U and du/dt, when
the acceleration becomes —

The increase of downward force is — pF2S . # Aa, while Aa = w/V.
Therefore —

. . w

The angular acceleration is rf?/^. One moment in this direction
is pF2Sc(rf&m/da)Aoc. Others arise in a complicated way, but the
most important of these is due to the increase of incidence of the tail
plane by IqfV, and amounts to — pF2S'/ . a'(lq/V) in the sense of a
increasing. Hence to the present approximation —

g dt •"
The two equations (i) and (ii) are conveniently written —

iik2 da w dkm

vci = T-£-™«- • w

From these follow two expressions for q :

_ 1 dw a __ 1 /w dkm yjtf dq\

q~~Vdt +~ylW~~™' \l T* "" Vc dt)9 (V)

Differentiate (iii) and substitute for dq/dt from (v), obtaining —

0 • • (378)

dP ' * A
where —

__ 1 /T0'c #\

(x \ Aua //

(vi)

C /T<W' ^^m\

v\ — — I .^^ _ !•

A similar differential equation may be obtained for q.

500 AERODYNAMICS [CH.

Substituting from w = w^in (378) gives the following equation
for the damping factor :

whence —

The form of the roots follows for a given craft at a particular speed,
enabling stability in the present connection to be examined.

If, for instance, YJ = 0, one root is zero and the other = — £•
The motion is then not an oscillation, but a stable subsidence or an
unstable divergence according to the sign of £.

Usually, however, 4v) > £2, when a pair of complex roots results,
which indicates an oscillation. Writing these roots —

X/F = - tf ± if,
— J£F is the damping factor and 2n/(}V is the periodic time.

268. Examples

The following particulars relate to a small, slow, lightly loaded
biplane at 100 m.p.h. :

M, c SIS' klc He T a a' dkm!da

6 5ft. 10 1-0 2-8 0-28 2 1-6 -0-12

From (vi) of the preceding article we find £ = 0-0378, 73 = 0-00062.
Hence —

X/F -= — 0-0189 ± 0-0162*.

The time t of a complete oscillation = 27c/(0-0162 x 146-7) =
2-64 sees. The damping factor = — 0-0189 X 146-7 = — 2-77.
The time to half-disturbance = 0-69/2-77 = 0-26 sec., or < 0-1 t.
Thus the oscillation is very heavily damped.

Let us next suppose that at 50 m.p.h. the aeroplane is approaching
its stall, and that a, a' are then reduced by 80 per cent. ; for simpli-
city dkm/d(x. will be kept the same. We find that t is doubled, while
X is reduced by 90 per cent., so that before 50 per cent, of the initial
disturbance is damped out, nearly half a complete oscillation is
described, or the craft travels 183 ft. If, at a still lower speed,
both wings and tail arrived at a flat stall, a damping factor would
cease to exist. The formula for J; shows, in short, that causes of
decrease in damping are : increase of (JL, i.e. increase of wing loading

XII] SAFETY IN FLIGHT 501

or altitude, or too short a fuselage ; increase of moment of inertia ;
decrease of tail volume ; decrease of lift-angle slopes.

We must not strain this approximate analysis too far in seeking
to apply it to practical flight. But the following conclusions which
it demonstrates are well established. The immediate response of a
stable aeroplane at high speed to a vertical gust or like disturbance
is an exceedingly rapid dead beat adjustment into the wind ;
damping becomes light at low speed near the stall, and an oscillation
may develop, of period usually less than 5 sees.

269. Lanchester's Phugoid Oscillation

In this article we assume the short oscillation to be damped.
Considering the effect of a transient upgust on initially steady
horizontal flight, the craft, turning almost instantly into the relative
wind as described in the preceding article, proceeds to fall and gather
speed. The resulting increase of lift provides an upward accelera-
tion, eventually stopping the fall and sloping the path upwards
again. The airscrew axis, which at first dipped, recovers and passes
through its original inclination to the horizon. Speed decreases
and, as the craft completes its small climb, becomes low, so that the
aeroplane must dive again to recover speed. Unless damped out or
corrected by the pilot, the cycle of changes repeats itself with a
period seldom less than 15 sees. This is called the long, slow, or
(after Lanchester) the phugoid oscillation.

For the present we shall be content to examine a greatly simplified,
or idealised form of the phugoid oscillation, following closely Lan-
chester's * original demonstration. The simplifying assumptions
are : constant incidence to instantaneous flight path, a propeller
thrust that always exactly balances drag, small size of craft com-
pared with the minimum radius of curvature of the flight path,
and negligible moment of inertia. Then the Aerodynamic force is
perpendicular to the path, and varies with the square of the speed.
But the angle 0 between the vertical and the normal to the flight
path is assumed to be sufficiently small for the difference between
cos 6 and unity to be neglected, so that the vertical component of the
Aerodynamic force is sensibly equal to the lift. It is implied that the
oscillation is of small amplitude, and it follows that squares of
velocity increments are negligible.

The motion is governed by an alternating exchange of potential

* Aerial Flight, v, 2 : " Aerodonetics," 1908. In this historic work Lanchester
proceeded to introduce corrections for moment of inertia, etc,, but such development
will be approached in another way.

502 AERODYNAMICS [CH.

and kinetic energy, which is conservative. Let the altitude of the
craft increase by h while speed increases by u. The K.E. increases
from WV'ftg to W(V + u)*/2g, i.e. by WVu/g, and—

Wh + WVu/g = 0. . . (i)

Owing to constancy of kL —

Lift (L) = W(V + w)VF» = W + W . 2u/V
or —

8L = L-~W=-W. 2gh/V* . . (ii)

on substituting for u from (i). Hence the vertical force increment
varies (sensibly) as the vertical displacement of the craft from a mean
level ; the oscillation is simple-harmonic with period —

F*\ V

..__) = 7^2 • - = 0-138 V. . (380)
2g/ g v '

Form of the Oscillation. — From (i), — u/h = g/V = constant,
whence the velocity variation is also simple-harmonic, 90° out of
phase with the vertical displacement. Let ut be the maximum
velocity variation and xlt ^ the semi-amplitudes of the motion of the
C.G. of the craft relative to axes moving uniformly with velocity V
and periodically coinciding with axes fixed in the craft. We have
xl = 1*^/27* or, since nx = — gh^V and from (380)—

Thus the superimposed motion of the C.G. is elliptical, the vertical
amplitude being V2-times the horizontal amplitude. This result

FIG. 189. — EXAMPLE OF PHUGOID OSCILLATION.

enables a phugoid oscillation of chosen length and vertical amplitude
to be traced ; an example is shown in Fig. 189.

The simple formula (380) indicates at 60 m.p.h., for example,
12-1 sees, for the time of a complete oscillation and 0-2 mile for its
length. This estimate would not be widely wrong for an aeroplane
of low minimum flying speed. But (380) would normally give much
too small values at, say, twice the above speed, when the oscillation
may exceed a mile in length. Since, in a real aeroplane, propeller
thrust oc IjV and drag oc Fa at constant incidence, damping of speed

XII] SAFETY IN FLIGHT 503

changes would evidently occur. Other damping arises from inci-
dence changes, here excluded. It is not necessary for the phugoid
oscillation to exist ; two subsidences may take its place at high
speed.

THEORY OF LONGITUDINAL STABILITY
270. The Classical Equations

As already mentioned, the theory of stability is founded on the
assumption that it is sufficient to examine the response of the craft
to small disturbances only from steady flight. We restrict in-
vestigation to straight flight, and in this case it was first remarked by
Bryan and can readily be verified that a generalised disturbance may
be resolved into components which affect the symmetric and asym-
metric stability of the craft quite separately. Thus we are able to
consider longitudinal stability without reference to lateral stability.
Considering an aeroplane climbing steadily at angle 00 to the
horizon, with Ox in the direction of motion, the conditions for
equilibrium are —

- g sin 0. + X. = 0

g cos 00 + Z0 = 0 . . (i)

M. = 0,

the components of Aerodynamic force being reckoned per unit mass.
The latter are different functions of U0f W0, the steady values of the
velocity components U, W. Obviously, q = 0.

Resulting from disturbance let U0, W0t and 00 increase by the
small quantities u, w, and 9, and let q appear as variable. Equations
(i) are no longer satisfied, accelerations appearing along Ox, Oz, and
about Oy. These are found, as in Article 267, to be —

dU/dt + Wq = duldt + Wq, dW/dt - Uq, dq/dt.

Each of the air forces, as well as the pitching moment, is affected
by uf w, and qt unless shown otherwise ; for the increase of X to the
first order we have —

Also the gravity components receive small increments. We write
Xu for dX/dU, Mw for dM/dW, and so on. In this notation the force
along Ox, for example, becomes —

- g sin 00 + X. - g cos 6. . 6 + uXu + wXw + qXq

504 AERODYNAMICS [CH.

and the first two terms vanish by (i). Hence we have the following
equations —

du/dt + Wq+gcos8Q.B^ uXH + wXw + qXq
dwjdt - Uq + g sin 0Q .6 = uZu + wZw + qZq . (381)
B . dq/dt = uMu + wMw + qMq.

It is to be noticed that we can substitute \qdi for 9 in these linear
differential equations. They are founded on the assumption that
u, w, q are so small after the impulsive disturbance that second order
terms may be neglected. Recently, this assumption has been found
to be insufficient for a peculiar reason in regard to the last equation,
and the addition of the term wM^ to be necessary, where
w=dw/dt. This will be discussed in detail later and an appropriate
correction introduced. We might also add to the last equation a
term representing an instantaneous movement of the elevator ;
a fourth equation might be framed for the propeller thrust.

Equations (381) determine u, w, q as functions of /. Any two
variables may be eliminated in turn in the usual manner, and a
differential equation of the same form results for u, w, or q, viz. —

f(D) . (u, w, or q} = 0,

where f(D) contains powers up to the fourth of the differential
operator. The solution of this equation is known to be of the
form —

u = u^ -f u^ + u^1 + w4£v,

where ult etc., are constant coefficients derived from initial values,
and Xlf etc., are the roots of the equation /(X) = 0. Since D^—'ke*,
D*e^* = X2^', etc., "ku may be substituted for du/dt, etc., whence
from (381) we find * for/(X) = 0 the equation—

-Xu -X. gcosda-l(Xq-Wa)

- Z. \-Za g sin 08 - X(Z, + U9)
— Mu — Ma X»5 — XM,

= 0

(382)

On expansion this equation may be arranged as :

A^ + BQK + C0X2 + DQ\ + E0 = 0 . . (383)

where the coefficients of X, called the stability coefficients, are func-
tions of Xu, Mq, B, 00, etc., all of which can be ascertained for a
given aeroplane under particular conditions. The response of the
aeroplane to small disturbance is investigated from the nature of the

* For a more detailed demonstration see Cowley and Levy, Aeronautics in Theory
and Experiment, 1918.

XII] SAFETY IN FLIGHT 605

roots of this equation, which may be real or complex. Stability
occurs if, as before, their real parts are negative. The algebraic
condition for this result is that all the stability coefficients — together
with Routh's discriminant :

Z~> ___ 75 /"* T"\ _ A T~\ J _ T^ 271^ /QQ*A\

are required to be positive.

The demonstration is completed with expressions for the stability
coefficients. It is convenient, however, to delay these until the
equations are expressed in more convenient units.

271. Glauert's Non-dimensional System

It will be seen that the stability equations are rather complicated.
Their discussion is greatly facilitated by adoption at once of the
dimensionless system of units due to Glauert.*

The quantities Xu, Mq, etc., are called resistance derivatives ; the
first named is further distinguished as a force-velocity derivative, the
second as a moment-rotary derivative ; to take another example, Xq
is called a force-rotary derivative. Inspection at once shows that
the derivatives depend on pFS (not on pF2S), S being taken for
convenience as the wing area, and we accordingly divide by this
quantity. Again, it is inconsistent to reckon forces per unit mass of
the craft, and yet to leave moments as actual moments. Finally,
most of the old derivatives are negative, and it makes for clearness
to change their signs.

Consider, for example, the force derivative Xu. When UQ changes
to f/0 + u> ^Q changes to X as given by —

X = A t

But, if m is the mass of the aeroplane, mX0 = — JcDpU02Sf provided
iD is calculated to include parasitic drag. Hence —

mXu - w-— ? - - 2fcDp[70S.

U Q

It is evidently suitable, therefore, to replace Xu by the non-dirnen-
sional coefficient xu defined (if W is the weight) by —

CD). . . (385)

Now Xu has the dimensions l/7\ Hence an appropriate unit of
time /0 in the non-dimensional system is —

tg =, W/gpVS. .... (386)

* Loc. cit., p. 381

506 AERODYNAMICS [CH.

Lengths must be expressed in terms of some representative length,
and the effective lever-arm / of the tail plane from the C.G. is chosen.

It follows that the unit of velocity is l/t0, and this = F/fi ft. per
sec. if —

p^W/gpSl .... (387)

as in Article 267. The significance of (z, the relative density of the
aeroplane, will already have partly appeared ; it collects the effects
of size, wing loading, and altitude. But the following is of interest.
Let us define &R, the coefficient of resultant air reaction, by —

jfcRpF2S = W cos 00: . . . (388)

For small values of 0Qt kR closely equals the lift coefficient, and, if
cos OQ = 1 —

F*S W F»

' ' ' ( }

The quantity on the R.H.S. will be recognised as the appropriate
parameter (cf. Article 66) for similar motions that are affected by
gravity.

A non-dimensional force-rotary derivative is obtained by dividing
that of the old system by pFS/, multiplying by m, and changing
sign, e.g.—

*, = _ mXJpVSl.

Finally, moments of inertia are expressed in the form —

B = kBml*,

the coefficient being called an inertia coefficient, and the moment-
velocity and moment-rotary derivatives of longitudinal stability
are —

272. Recast Stability Equation

In terms of the non-dimensional system the equation (382)
becomes —

JI ~~

= 0
^' (391)

XII] SAFETY IN FLIGHT 507

This is simplified further by (a) introducing wind axes, (b) neglect-
ing the derivatives xg, zq, which are always small. The result is —

= 0
. . (392)

** X + ** fife tan 00 - X)
™>u mw X» + *mq

Expansion gives —

K + Bjt + Cjf + D^ + Ei^O . . (393)
in which —

mq(xu + zw) + \unw [ == V^ + pmw, approx.]
DI = *»,(*A - *» ^) + ^{^K ~ ^R tan 60) - mu(xw + kR)}
EI = H'M^fo — ^u tan ff0) — mu(zw - xw tan 00)}
[ = V>kL(zumw — zwmu), approx.] ..... (394)

For stability, all the coefficients and Routh's discriminant must
be positive.

273. These criteria for stability, though greatly simplified, are
still rather involved. But the following may be noted at once.

It is generally true that Dl and El are small compared with J5X and
Clt enabling (393) to be factorised approximately as —

(XB + £lX + CJ X" + -ll ~ll X + ~ 0 . (395)

The first factor equated to zero gives the short oscillation (cf. Article
267), the second the phugoid oscillation or the subsidences which
may take its place. Now, Routh's discriminant has to do with
oscillations. Unless an oscillation increases in amplitude through
the discriminant becoming negative, instability must first appear
(in a nearly stable aeroplane) through El becoming negative. To
the approximation in (395) the conditions for stability are —

El and (C1D1 — B^E^) both positive. . . (396)

The first of these conditions means mw positive, in horizontal flight,
and so refers to the investigation of Article 266.

It is seen that [JL is always associated with mw or mu, in which it
might be included. This usefully localises effects of wing loading and
altitude.

Since the scale of time is changed, the new damping factor and the
new period of oscillation are (if the accent refers to the old) —

X=X'*pf t = f/t. . . . (397)

508 AERODYNAMICS [CH.

and we have, if or is the relative density of the air —

= 0-637

ENGINE-OFF STABILITY

274. Force Derivatives

Certain practically useful formulae will now be obtained that are
restricted to normal flight incidences with engines off (often known
as glider stability).

In steady flight with wind axes — mX = D, the total drag, and
_ mZ = L, the lift. But angle disturbance causes the directions
of X and Z to cease to coincide with those of D and L. If incidence
increase by the small angle w\V ', we have —

ze> ...

IS)

- mZ - L + - D. . . . (ii)

Z and X, called the normal and longitudinal forces, respectively, can
evidently be plotted from lift and drag data, and slopes at the un-
disturbed incidence measured. But we can calculate approximately
as follows. By differentiation of (i) —

~~ 8w\ V ) ~~" V\da /
whence —

This result is simplified by the substitution, appropriate to moder-
ately small incidences (cf. Articles 176, 247) : £D = &<> + kJcL*t
giving dkD/d<z = 2kJcLa, a denoting dkjda as before. k0 will
include parasite drag.

Proceeding in this way we find —

#u = 2kD - 2(fc0 + kJcL*) (=CL)

(398)

zw = a + &0 + &1&L1 = a> approx.,
while it will be remembered that xq = zt = 0, approximately.

XII] SAFETY IN FLIGHT 509

275. Moment Derivatives

One moment, mut vanishes in gliding flight. Another, mw, is
easily determined by experiment, or can be calculated roughly, in
favourable circumstances, on lines indicated by Article 266. If
Tcm is the coefficient of resultant moment, we have —

_ _ Mw __ 1 9M __ c dkm
kBpVSl ABpF25/ 3oc kBl do. '

and now if only wings and tail contribute, we have clearly, from
Article 266 —

m™ = Fz IT x1 ~~ d~) ~~ ba\ ' ' * • (40°)

But in regard to modern high-speed monoplanes, this formula suffers
from the restriction mentioned at the end of Article 266.

The remaining derivative, mgt must be obtained from experiment,
unless correction factors are available from experience, and presents
a difficulty which leads to the introduction of another derivative,
m^, as anticipated in Article 270. These are discussed in the next
article.

276. mq and m^.

The basic data for mq are obtained by oscillating in a wind tunnel
a more or less complete model of the aeroplane concerned about the
transverse axis through
Two methods
The model

its C.G.
are in use.

may be oscillated freely
by means of a spring and
the logarithmic decrement
of the damping estim-
ated by measuring the
amplitudes of successive
swings. Mechanical fric-
tion accounts for some of
the observed damping,
but is allowed for by
repeating the experiment
in still air. In the alterna-
tive forced oscillation

method, the model is oscillated through a spring and wire by a
crank (Fig. 190) ; the frequency of the applied oscillation is

FIG. 190. — AN ARRANGEMENT FOR MEASURING
AIR DAMPING,

510 AERODYNAMICS [CH.

gradually increased until the model attains a maximum synchronous
vibration, the amplitude (0J and period (t) of which are observed.
If v denote the damping due to the wind, Vj that due to friction, and
0! the amplitude of the forcing oscillation, it is nearly true that —

v + vt = A.|. . . . (401)

vx is extracted, as before, by repeating without the wind.

Either of these methods determines v satisfactorily. But it does
not follow that the air-damping is the same as Mq ; in fact, these
two quantities are essentially different from one another. The
reason is as follows. Mq, like other derivatives, must be evaluated
with other variables kept constant. But in the experiment a varies
as well as q. Moreover, the time 8t taken by a change of downwash
8s due to change of wing incidence to travel from wings to tail is so
considerable that the tail meanwhile changes its incidence appreci-
ably. Now Mq results almost entirely from the tail, and downwash
decreases tail incidence, which is increased, therefore, by the lag.

During $t let incidence increase by Soc. With the approximation
& = ljVt we find Sa = a (l/V) and tail incidence is increased from
Iq/V to—

Iq ds Iq dz . I I

since q = da/dl. Thus the efficiency of the tail is increased by the
factor within the brackets, and approximately —

i.e. the measured air-damping is to be decreased by about one-third.
Experiment aifords support of this conclusion.* Finally,

' (403)

Experimental determination takes account of interference from
the wings (due to wake velocity reduction) and from the body, and
includes damping from parts other than the tail represented in the
model. If we neglect small contributions, and can introduce a
factor / for interference, we calculate as follows. We have —

= ^-

* Aerodynamic Theory, vol. v, p. 51.

XII] SAFETY IN FLIGHT 511

whence —

<mq = / r-rTfl1 .... (404)

No downwash factor appears, because incidence is assumed constant.
It remains to deal with the remainder of the air-damping, v, i.e.
the last term of —

v = Mq + Mq(de/d«)

which follows from (402). Since q = d and a (when q = 0) = w/V,
a = w/V and —

1 1 de

Hence —

= mq . (rfe/rfa) ..... (405)

The effect on the stability equation of including this additional
derivative is to add the term 'hm^ to mw in (392).

277. Example

Further discussion will be illustrated by reference to a monoplane
of wing loading 13-8 Ib. per sq. ft. at 150 ft. per sec., to which the
following particulars apply :

H dkj r a a" b k^ *D *t
12 3 J 2 1 0-075 0-258 0-025 0-1 0-35

It is assumed for simplicity in this example that formulae (400) and
(404) without an interference factor are suitable, provided a" is
substituted for a' . Hence —

mw = 3(0-333 X 0-65 — 0-15) = 0-2 ; mq=l.

We also have to a close approximation &R tan 00 = — kL . kD/kL ==
— &D, while mu = 0 in gliding flight.

Neglecting m^t we find for the approximate stability coefficients
of (394) :

Bl = 2kD + a + mq = 3-05,

C1 = amq + \wiw = 4-4,

Dl = mq{2k»a - 2^(2^ - 1)} + 3jxfcDww == 0-36.

El = 2i>.klmw = 0-32.

612 AERODYNAMICS [Cfi.

All the coefficients are positive. Routh's discriminant —
B^C-iPi — ZV — BI*EI = 4-83 — 0-13 — 2-98

is also positive, and the craft is stable.

Short Oscillation. — From (395) this is given by —

X* + 3-05X + 4-4=0
or —

X = — 1-525 ± \ A/9-30 — 17-6 — — 1-525 ± 1-44*.
The period is —

t (sec.) = — — tQ = 5-2,
v ' 1-44 °

because tQ = 13-8/^pF = 1-2. The time to half-disturbance is % sec.

278. The Long Oscillation.

To the approximation of (395) the period of the phugoid is —

t (set.) = 27tf0VC1/£1. . . . (406)
nearly. For the example this gives —

t (sec.) = 2-47tV'4-4/0-32 = 28.
Similarly, the damping factor in common units is —

Li~ii ^407)

to' 2CV • • - , ;

evaluating to —

- J_ X 4±X 0-36 ~ 3-05 XOJ2 =
1-2 2 X 19-36

and the time to half-disturbance is 0-69/0-0131 = 53 sees.

It is of interest, however, to develop approximate formulae in
terms of the derivatives for gliding flight. Substituting in (406) —

Luw

^Y . . (408)

This result should be compared with (380) ; that simple and pre-
liminary estimate of the period is to be increased by the factor under
the radical. A rough formula for the second term of this factor in
the case of a monoplane having unimportant pitching moments
from the body and engine nacelles is easily obtained from (400) and

XII] SAFETY IN FLIGHT 613

(404) as (0/(ji)/(0-65 — fib), where /3 will be of the order 6. Thus,
considering a craft of given weight and shape, moving the C.G. back
may soon produce a long period. If mw vanish, no oscillation will
occur. Then E± is no longer positive, and the motion is a weak
divergence. If, on the other hand, the C.G. is far forward and ntw
large while (owing to small tail volume) mq becomes small, a large
moment of inertia is likely to make the oscillation increase in
amplitude.

SOME STALLING AND ENGINE-ON EFFECTS

279. The foregoing analysis may be employed for gliding flight
up to 8° or 10° incidence, but errors then begin to become appreci-
able. The method ceases to be useful near the stall. Nothing
serious happens to the phugoid, as a rule, on reaching the stall, but
the short oscillation may become unstable through a narrow range
of speed. Effects primarily resulting from large decreases in a and
a', and from reversal of sign of the former, will easily be followed
out. Some notes are given below on other effects.

Tail Level. — The efficiency of a tail plane is reduced by the wings
on account of their downwash and wake. It is not feasible to
obviate the first loss except by ' tail-first ' layout. The second may
involve a factor so low as 0-75 through decreased air speed if the tail
plane is directly in the wake, but can be avoided at small incidences
by carrying the tail plane either high or low. Now wings which
approach a rectangle in plan-form stall first near the centre of span,
several degrees earlier than the incidence for maximum lift. Down-
wash then decreases at the tail
plane, while the wake thickens
and lifts — two changes which
are conflicting in regard to
tail efficiency. The high tail
plane suddenly becomes poorly
situated at large incidences
through passing into the wake,
decrease of downwash failing
to compensate for loss of
speed ; the tail plane which is
normally level with the wings
may run through the wake and
become more efficient. These
effects are illustrated in Fig.
191.

10° 20°

Incidence

FIG. 191. — EFFECTS OF TAIL LEVEL ON
ix$ EFFICIENCY.

A.D.— 17

614 AERODYNAMICS [CH.

With wings having very sharp taper, stall may set in from the
tips * and the above transition be delayed. Sharp taper is also
associated with a concentration of downwash behind the central
part of the span. Tail location for biplanes is governed by the fact
that the lower wing usually stalls late.

It will be deduced that a rather low position for the tail plane is
usually preferable from the present point of view. However, this
frequently tends to increase parasite drag and to introduce landing
difficulties.

280. Effects of Stalling on Moment Derivatives

In regard to miv, most aeroplanes have such strong static stability

at large incidences that elevators
of normal size soon fail to be able
to depress the tail further.

Variation of mqt or mq + m^,
normally takes the form illus-
trated in Fig. 192. Near the
stall, damping passes through a
sharply defined minimum, which
has little to do with the tail,
but is due essentially to rapid
changes in the pitching moment
of the wings. Instability may
result from this cause through a
narrow range of low speeds.
First approximations to the
damping factors of the short
and long oscillations are, respectively, mq and a becoming small —

— £K + a) and — 3*W2-

The large increase of drag at stall tends to keep the phugoid stable,
and the trouble is seen to be concerned with the short oscillation,
normally so strongly damped. The question is intimately concerned
with the shape of the lift curve at the critical angle, i.e. on the
violence of the stall. This is less severe if burbling creeps forward
from the trailing edge than if, as happens with some aerofoil shapes,
it sets in near the nose.

38 1. Level Flight

It is by no means certain that an aeroplane which is satisfactory
when gliding will maintain stability when the engines are opened out
* Nasir, Jour. R.Ac.S., August 1935.

0°

10°

Incidence

20°

FIG. 192. — VARIATION OF mq WITH
INCIDENCE (TYPICAL).

XII] SAFETY IN FLIGHT 516

for level flight. The chief differences that occur are summarised
below ; further study may conveniently proceed by a method of
graphical analysis which will be described later.

The first change to note is that 00t = — tan ~ l (&D/&L) during
gliding, vanishes, thereby reducing the stability coefficient Dt.

Several important changes affect the tail plane, assumed to lie in
the slipstream. The first is an increase of downwash, which is
difficult to calculate as local changes of trailing vorticity are in-
volved ; at present this is best estimated from special experiments
(the cautionary note may be made that increases are often surpris-
ingly larger than would be indicated by simplified theories). The
chief result is a decrease of static stability through modification of
mw. Secondly, the tail plane is subjected to an increase of speed,
which may be calculated by the methods of Chapter XI, and has the
effect of increasing mq. Thirdly, we note that the slipstream will
vary with small changes of V due to variation of propeller thrust T,
and that mu will take up values in consequence. It is not difficult
to estimate dT/dV from Chapter X, constant engine torque being
asstimed for this purpose. But if the speed at the tail plane increase
from V to V\/r, a formula for mu is —

•*-£-7-2P* ' ' ' (4°8)

Besides its effect on mu, dT/dV directly modifies the gliding
formula for xu to j-p

' ' • (410)

It should be observed that &D is not the same in this formula as in
that for gliding, because of the elimination of the idle airscrew drag
of the latter case. For this reason, xu is often much the same in the
two cases.

282. High Speeds

On resolving (407) in terms of derivatives, a long expression results
for the damping factor of the phugoid oscillation. This at once
simplifies, however, if kL become very small, corresponding to high
speeds. Damping is then found to be proportional to kD as a first
approximation, a result that may be compared with a remark at the
end of Article 269. It is somewhat unfortunate that stability, from
the present point of view, decreases as the craft becomes more
efficient.

Putting a = 2, the approximate expression for El is —

616 AERODYNAMICS [CH.

On kL decreasing, the first term loses importance, and stability, as
depending upon the sign of El9 becomes more and more determined
by that of mu. mu may change sign as speed attains high values and
so produce instability. Possession of static stability does not guard
against this eventuality.

283. Free Elevators

It is desirable, chiefly in order to reduce fatigue during long flights,
for craft to be stable when elevators are released. The foregoing
methods suffice to investigate this question in a given case, provided
allowance is made for loss of efficiency by the tail plane. Following
disturbance, free elevators will normally change their incidence by a
less angle than if they were fixed, moving relative to the fixed part
of the tail plane until the moment about their hinges again vanishes.
The loss in tail efficiency may be as great as one-third. However,
suitable balancing or springs can obviously be arranged to prevent
the loss, or even to increase stability with free elevators by causing
them to move through a greater angle than the craft on disturbance.
This is a principal consideration in favour of minimising the chord of
elevators.

284. Climbing

Referring to (394), climbing — 00 positive — is seen to diminish
DI and Et. The latter effect is usually negligible, and hence the
result is to decrease the damping of the phugoid. The amount of
the decrease is approximately [MnwkL tan 00/CV

285. Graphical Analysis

An ingenious and rapid means of examining stability by graphical
means has been devised by Gates.* If we define —

Xo^W1/*1* yo = ™/.c//i« . . (412)

where k is the radius of gyration, so that X0 — 0 is the Aerodynamic
centre, we find that mw and mq will depend on XQ and Y0, but the
other derivatives on kL. The question of stability can be exhibited
by plotting curves against XQt Y0 as co-ordinates, kL and (JL being
supposed constant. The composite curve : Routh's discriminant
jRt = o and E1 = 0, will represent a dividing line or boundary
between stability and instability.

If, for example, we assume (400) and (404) to hold without an

* A.R.C.R.&M., 1118, 1921

XII] SAFETY IN FLIGHT 517

interference factor for a particular monoplane in which a = 2»5 =
lie and de/d* = 0-35, we have, during gliding :

mg = Y0, mw = 0-65 Y0 - X.

and #! = 0 gives X0 = 0-65 Y0, independent of kL and JA. But
Rt = o clearly depends on &L and p.

Fig. 193 illustrates the form of the boundary curve for gliding
flight, as varying with
kL. The broken line in-
dicates the approxima-
tion to R! = 0 given in
(396). Crossing £\ = 0
towards the right in the
figure means that a di-
vergence occurs, the
curves being shaded
towards the stable region,
and both C1 = 0 and
Dl F= 0 lying to the right of El = 0. Bl = 0 lies far below the
figure, parallel to the X0 axis, and the possibility of #! becoming
negative need not be considered. Crossing Rl = 0 downwards
signifies that an oscillation increases in amplitude. It will be seen
that this eventuality is much less probable at high speeds than at
low. Fig. 194 shows the effects of increasing ^ at a low speed (kL

0 X0

FIG. 193. — TYPICAL STABILITY CHART.

FIG. 194.

constant and fairly large) ; with a high wing loading, especially
at great altitude, El = 0 approximates to the complete con-
ditions for stability. These illustrations are based on diagrams
given by Gates,* who also obtains typical boundary curves for
level flight, showing in particular a restriction of the stable region
for high-powered craft.

* This and some other writers on stability employ k» to denote the radius of
gyration about the transverse axis, instead of k as in the above. Reservation of this
symbol for the inertia coefficient : jfca//2, has been adopted in this chapter for ease of
reference to Jones's account of the subject, he. tit., p. 493.

A.D.— 17*

518 AERODYNAMICS [CH.

LATERAL STABILITY
286. Introduction

From a mathematical point of view, the conditions for asym-
metric stability are, with little error in normal flight, formally the
same as those for longitudinal stability. The quite different rela-
tionships between corresponding derivatives, however, change the
physical aspect completely.

The aeroplane is conceived to be flying straight in its plane of
geometric symmetry when lateral balance is disturbed. The
asymmetric motion that results comprises three responses, which
develop at such very different rates that we can follow them in turn.
If a vertical gust strike a wing-tip a rolling about Ox will first occur.
The moment damping this formed the subject of investigation in
Article 93, being found to have a large magnitude except near or
past the stall. It is associated with the derivative Lp. Thus,
normally, the rolling subsides extremely rapidly, leaving the craft
with slight roll (list) and yaw. The sideslip that ensues generates a
rather complicated lateral oscillation — the only oscillation that
arises from asymmetric disturbance ; its period may be 5 or 6 sees.,
and, with a stable craft, it may diminish to half-amplitude in about
double this time. The craft is then left with a sluggish spiral
motion, associated with sideslip and yawing, which is slow to develop
or to decay.

Since the rolling subsidence is inherent in all aeroplanes at normal
incidences, asymmetric stability is concerned with the oscillation
and the spiral motion, and these depend largely on the disposition
of the vertical fins. The effective fins of primary importance are (a)
the actual fin together with the rudder, usually situated above the
tail plane, (b) the transverse dihedral angle of the wings. The
significance of the latter as a fin was seen in Article 95, from which
(or otherwise) will be apparent that fins in general lead to both roll
and yaw. Thus arise the derivatives Lv and Nv. Secondary fins
include the body, engine nacelles and airscrews. These are by np
means negligible, especially in the case of high-speed monoplanes of
modern types, (a) and (b) alone, however, are at the designer's
disposal in regard to stability ; they are adjusted to take secondary
fin effects into account and the latter will be omitted from discussion
for clearness.

It is important to realise that fore-and-aft balancing of fin surface
is critical. A craft left with positive roll sideslips to the right, and the
equivalent yaw produces lateral forces on the fins. That on the rear

XII] SAFETY IN FLIGHT 519

fin (comprising the actual fin and the rudder) turns the craft into a
right-hand turn suitable for the existing bank. The left-hand wing-
tip now moves with excess speed, and asymmetric lift may increase
both bank and sideslip. The resulting spiral flight is seen to follow
too strong a directional stability in the static sense. On the other
hand, much too far forward a position for the C.P. of the lateral
forces — strong directional instability — leads rapidly to a fast,
spinning dive. No definite result can be drawn from these simple
considerations ; it is not clear that any static directional righting
couple is desirable, slight directional instability of the static kind
often proving an advantage dynamically.

The above illustrations tacitly assume a weak or absent transverse
dihedral. As we have seen, a dihedral rolls a craft away from the
sideslip and turn. The outer wing-tip of the turn thus comes to be
at thq lower level, though its greater speed soon raises it again.
Reversal of the sideslip leads to the lateral oscillation. This would
not occur in a craft that was prone to spin. The latter defect being,
however, uncommon, the oscillation is usually present, and may, on
occasion, become noticeable, when sufficient (though not too great)
static instability exists, together with an exaggerated dihedral.

287. The Asymmetric Equations

Mass distribution is now defined by A and C, the transverse and
directional moments of inertia. For precision we have also to
include E, the product of inertia along Oy, though terms containing
this as factor may be neglected in normal flight. The gravity force
is in the direction Oy and amounts to g cos 00 . (f> per unit mass.
Construction of the classical equations for small oscillations follows
precisely the lines explained in Article 270 for longitudinal distur-
bances, and the reader will have no difficulty in verifying the following
group in place of (381) :

dv/dt + U,r -W0p-g cos 0, . $ = vYv + pYp + rY,
A . dp/dt - E . dr/dt = vL9 + pLp + rLf . (413)

' C . dr/dt - E . dp/dt = vNv + pNp + rN,

in which, since p = d<f>/dt — sin 00 dfy/dt and r = cos 00 dfy/dt while
d<f>/dt = X<£, we can substitute for $ from —

<h y

cos 00 . <£ = £ cos 00 + r sin 00. . . (414)

A A

Treating these equations in the same way as (381) gives for the
damping factor X the equation —

520

x-y.

**v

-N..

AERODYNAMICS

XM— XL,
-KE— X2V

-£sin00-X(Y,-C70)
— XlE —XL,
X*C-->JV,

[CH.
(415)

Expressed in the non-dimensional system, this equation can be
arranged in the form * —

E _ / W ,\

-x'-K

z, ,/ W«

*V (Jt*R — X f ^ — |x —

nf AR tan 00 — X (^f + (i -^

(416)

Away from the stall, terms containing E as factor may be neglected.
Comparison with (391) then shows formal agreement to exist accord-
ing to the scheme :

z
m

•y,

by which is meant that nr takes the place of zw9 — yp that of xqt etc.

288. Solution with Wind Axes

Introducing wind axes as in the longitudinal case, so that C70 = V
and W» = 0, and taking approximately E — yp = yf = 0, (416)
reduces to —

XnA

which expands to —

X4 + £2X3 + CtX' +

where, writing kL as an approximation to kR —
Bt = lp + nr+ yv (= lpt approx.)
C, = (/^fir - l,np) + yv(lp + nr) - tin,
£>t = %(/A - 4^) - jinw(^ - iL tan
£» = P*L{ft*r - W - tan 00(/^ - ^«,)}.
These expressions should be compared with (394).

* Glauert, loc. cit., p. 493.

— kR tan S0)

= 0

X/f

(417)

Xf + Xwf

+ £a = 0 . . (418)

(419)

XII] SAFETY IN FLIGHT 521

The aircraft is stable if the stability coefficients and /?§ = BftDt
— ZV — BfE* are all positive.

289. Approximate Factorisation

Arising from the overwhelming rolling subsidence there is in
normal flight always one large root of (418), viz. — B2 = — lp,
approximately, enabling the factor (X + lp) to be divided out.

Associated with the spiral disturbance a small root occurs, closely
of value — EifDi, which can also be extracted.

There is left a quadratic, representing in usual circumstances the
lateral oscillation. To a first and rather rough approximation this
is —

Xi + 7?x + 7ri=s0- • • • (420)

#2 #2

The magnitude of the damping factor calculated from this approxi-
mation errs on the wrong side for safety. If the quadratic has real

roots, it represents the spinning divergence.

•

290. Discussion in Terms of the Derivatives

The condition for Et to be positive in level flight is —

lvn,>l,nv .... (421)

and instability is usually traced to a failure here. The first two
derivatives are usually positive, the third considerably larger and
negative, so that av, if negative, must be small. This means that
static directional stability must be limited, though it is to be re-
marked that increase of — nv is accompanied by increase of nf.
More generally, the above condition is —

lv(nr - np tan 00) > n9(l, - lp tan 0f) . . (422)

which intimates that stability is a little more difficult to secure
during climbing.

The quadratic (420) is easily investigated once the stability co-
efficients have been evaluated in a given case. That it should have a
pair of complex roots will be found to depend on nvt if positive, not
exceeding a small fraction of /„. This value must be reduced con-
siderably if the oscillation is to decay quickly, though it need not
become negative, i.e. static directional stability may not be neces-
sary.

Change of sign of *lp near the stall has consequences traced in
Article 93.

522 AERODYNAMICS [CH.

291. Example

Some plausible numerical values are —

^ *L '» lp lr »v ** nr Vv

8 1- t 6 -2 -I 1 Va i

These give for level flight —

J5, C, /;, *.

6-46 5-79 15-38 -0-67

The craft is spirally unstable, to correct which a larger dihedral
would be required. Routh's discriminant is —

6-46 X 5-79 X 15-38 — (15-38)2 + (6-46)' X 0-67,

so that the oscillation is damped. Inserting values in (420) we have
approximately —

Xa + 0-9X + 2-4 = 0
or —

X = — 0-45 ± 1-45*.

The true damping would be appreciably less than as here estimated.

292. Evaluation of Derivatives

The only effective force derivative, %, is easily deduced from a
tunnel experiment in which a complete model is yawed at a succession
of small angles to the wind.

The strip method, introduced in Article 93 and employed in the
Theory of Airscrews, is available for the approximate calculation of
moment derivatives which depend largely on the wings. It is readily
arranged to take account of non-uniform grading of air load along
the span in steady flight. This is best seen from an example.

Let lr be required at a mean lift coefficient kLQ. Neglecting
contribution from a possibly high fin, consider two wing strips of
chord c distant ± y from the longitudinal axis. Owing to change of
local speed only, there arises an element rolling moment :

8L(f) = p<%fcLy {(V + ryY - (F - fy)1} = 49VrkLcy*ty.
If c and &L may be assumed constant, this integrates at once to —

= f

where s is the semi-span and S the total area. Otherwise, consider-
ing the integral —

we note that —

= c M
^Jo

XII] SAFETY IN FLIGHT 523

whence, if yQ be the radius of gyration of the lift-grading diagram
appropriate to the plan form and sections in steady flight,

L(r) = 2pFriLOSy;.
Finally —

~ ~~"W ~~ ~~ Wfe^

In the absence of precise data, elliptic loading may be assumed, for
which y0 = |s.

Rotary derivatives can be determined experimentally by the
oscillation method (Article 276)* or, in the case of lp and np, by
measurements during continuous rotation of the model about the
wind direction. nf depends principally on the fin and rudder and is
analogous to mq.

CONTROLS

293. The functions of Aerodynamic controls in steady flight under
various conditions have already been described. The subject of
aerobatics is beyond our scope. We now discuss the requirements
and limitations of controls during disturbed flight, examining the
first principles of their design in connection with the dynamic
features of the craft. Several preliminary considerations are
grouped together in this article.

Power. — In deciding the power for an Aerodynamic control, we
note first that the size of elevators, given the leverage, determines the
maximum angle of incidence of flight. These might be so dimin-
ished as to render impossible the deliberate stalling of the craft.
Such a policy has adherents on the Continent, but opposed to it is
the consideration that small elevators deprive the pilot of the means
of quick recovery from accidental stall, caused, for example, by the
sudden failure of an engine with a high thrust line. The ailerons,
having a strong adverse rolling moment to overcome at normal
speeds, must be many times as powerful as the rudder, for little
resistance opposes yawing. Use of ordinary ailerons induces an
adverse yawing moment, i.e. one which turns the craft to the wrong
hand for the imposed bank ; this is natural, since the wing with the
greater lift exerts the greater drag. The yawing is corrected by use
of the rudder. The necessity for this correction may be overcome
by a spoiler operating on the depressed wing, but synchronous rudder
movement appears to have become instinctive with pilots.

* Relf, Lavender, and Ower, A.R.C.R. & M. 809, 1921.

524

AERODYNAMICS

[CH.

(b)

The hinge moment of a control surface should be reduced by
balancing to small magnitude, determined by the force a pilot can
exert on the control column without fatigue. This balancing is
Aerodynamic and will not be confused with mass balancing, intro-
duced to prevent elastic flutter. It is achieved by locating the
hinge line well aft of the effective nose of the control surface. There
is some danger of over-balancing when fine limits are attempted for
a very large craft ; moreover, balancing commonly fails at some
moderately large control angle. For these and other reasons, the
control surfaces of heavy aeroplanes are often servo operated,

which usually means that the control
column moves a tab, or diminutive surface
attached to a control, in the wrong
direction, and the tab generates suffi-
cient Aerodynamic moment to operate
the real control. There are two general
points to notice. Aeroplane controls are
' reversible/ the pilot feeling a moment
proportional to that applied. He- is
usually able, from the point of view of
strength, to operate controls quickly
through wide angles.

Stalling of Controls. — It is clearly
important for controls to stall later than
the wing. Consider, as one of many
examples that might be chosen, the case
of an aeroplane which is approaching a
confined landing space steeply with its
wings at large incidence ; a wing-tip that
suddenly rotates downwards through a
gust is momentarily at still greater inci-
dence and yet must be lifted by its
aileron. The remarkable efficiency of
Handley Page slots both in delaying
aileron stall and in curing adverse yawing

moment at high incidences has been described in Article 94. Stall-
ing of a control is also delayed by the so-called ' cut slot/ which
may be arranged to open between a deflected control and the fixed
member to which it is attached. Fore-and-aft location of a cut
slot and also its shaping need care ; this is work for the wind tunnel,
but experience is necessary to allow for scale effect on both
variables. Slots should be closed when not in use.

•--:

(c)

FIG. 195. — (a) FRISE TYPE
AILERON ; THE NOSE ACTS
AS A SPOILER ON THE DE-
PRESSED WING ; C, CUR-
TAIN OR SHROUD, (b) HORN
BALANCED AILERON, (c)
OUTRIGGED SERVO (S)
OPERATING THE PARTLY
BALANCED RUDDER OF A
LARGE CRAFT.

XII] SAFETY IN FLIGHT 625

Rudder and elevators lose efficiency when the wings stall.
' Shadowing ' by the wings may be so marked during a spin that the
rudder is commonly set as high as structural considerations allow.

294. Relation to Stability

In normal disturbed flight, controls are used to correct a tedious,
though safe, response of a stable aeroplane and also to supplement
its stability if this be restricted.

Elevator control cannot vie with the speed of the short, damped
oscillation — a chief reason for limiting longitudinal static stability,
excess creating a discomfort in bumpy weather which is beyond the
power of the pilot to ease. If the oscillation becomes unstable near
the stall (Article 280), there results a lag in the response of the craft
to the elevators, making steady flight difficult or impossible. Special
modifications to controls have been suggested.*

The sluggish phugoid can be corrected at an early stage.

Turning to asymmetric disturbances, there is ample time to
correct by control movements both the lateral oscillation and the
slow spiral motion. But it is doubtful whether a pilot can act
sufficiently rapidly to prevent failure by the spinning divergence, if
this supersedes the lateral oscillation, especially as the cause is likely
to be too small a rudder. It is noteworthy that since the spinning
divergence may double disturbance every two seconds, whilst spiral
instability, due to too great static righting moments, develops only
very slowly, there exists a mathematical explanation of the pilot's
well-known distrust of a weak rudder. Possession of too powerful
a rudder prevents flying with all controls released, yet nearly all
aeroplanes are so characterised, and it is probable that spiral in-
stability will remain a general feature until amateur, fair-weather
flying greatly increases.

In favourable circumstances, a stable aeroplane will fly itself just
as steadily as a pilot can contrive ; it is equally true that an unstable
aeroplane can be flown satisfactorily with skill and good controls,
provided it is not wilfully unstable. An important aspect of stab-
ility is to provide relief for the pilot, and this will not be necessary
during manoeuvring. Thus a craft may be stabilised for level and
gliding flight, but left to control during climb.

The investigation of this chapter is restricted to symmetric flight,

and an aeroplane designed in accordance with the theory given might

become unstable if, for example, it were turned into horizontal

circling flight, when the damping is redistributed between the short

* Garner and Wright, A.R.C.R. & M., 1193, 1028.

528 AERODYNAMICS [CH.

and long oscillations. Extension of the theory is possible, but
simplicity is lost through coupling of longitudinal and lateral effects.
If it be thought that so restricted a study of the subject loses practical
utility, it should also be realised that knowledge of a craft's symmetric
stability will often suggest a modification to correct faulty behaviour
in a manoeuvre.

295. Large Disturbances

Experience has shown that most craft, when subjected to common
disturbances, behave in much the same way as calculated by the
theory of small oscillations. But guidance through very large
disturbances must be left to the pilot ; it is just in these circum-
stances that he stands most in need of unfailing controls ; and the
obvious deduction is that their efficiency should be ensured under
extreme conditions. Aeroplanes are controllably safe in this wider
view except as regards recovery from a possible type of motion
known as the flat spin.

The flat spin may arise as a development of the slower spin
described in Articles 92-94. When this ordinary spin speeds up and
narrows, large centrifugal couples tend to lift the nose of the craft,
and in some cases to increase the rate of spin rapidly. At a large
incidence, in the neighbourhood of 45°, the elevators may not be
able to check this tendency. The rate of descent then decreases,
but the rate of spin increases from perhaps 20 r.p.m. to 60 r.p.m.,
and the only chance of control lies in the rudder. Detailed analysis
must be left to further reading, but has not yet proceeded very far.
An excellent experimental method of investigation is provided by
the free spinning of light models in a vertical tunnel (cf. Article 66),
although the usual difficulties exist in carrying over such small-
scale experiments to full scale. The distribution of mass in the craft
appears to be of great importance.

THE LOAD FACTOR

296. Whilst structural design is no part of our subject, Aero-
dynamics specifies largely what load each member of the craft
should be able to withstand. The possibility of large transient
accelerations makes a wide margin of strength especially desirable in
aircraft, but the imperative need to save weight must narrow this
down to the safe minimum. An over-all factor of safety, such as is
commonly used in engineering design, would obviously be wasteful,
and, as a rule, the maximum load that each part of a craft is likely

XII] SAFETY IN FLIGHT 527

to be called upon to withstand is carefully assessed before applying
an over-riding factor to allow for defects in design, material, and
workmanship. It is unnecessary to consider the matter in detail,
since rules are laid down by the competent Government authority,
which take account of the duties which a given craft will normally
discharge, and we shall consider only some principles.

Instances have appeared in Chapter IV and elsewhere, e.g. during
steady or unsteady turning, where clearly the wings support a load
equal to several times the dead weight of the craft. The ratio of
this load to that supported in straight level flight is called the load
factor. Considering all cases that may be specified by the accelera-
tion of the C.G., we easily find that the load factor is the ratio of the
abnormal to the normal lift coefficients, change of speed being
neglected, and that its maximum value is the ratio of the maximum
lift coefficient to the normal. The critical condition arises in pulling
out from a steep dive, when kL is initially very small, and simple
calculations show that the load factor might then exceed 50 if the
elevators were suddenly operated by radio from a distance. But a
personal load factor of 5 would cause acute physical discomfort to a
pilot, who therefore straightens out much more gradually. Thus the
pilot acts as a safety valve against excessive load factors.

The accelerometer records the variation of load factor during flight.
This instrument may consist of a short fine wire clamped at the ends
like a beam to a frame secured to the aeroplane and with its length
perpendicular to the direction of the acceleration to be measured.
The wire bends under its own weight and centrifugal force, and the
deflection is photographed on a moving film. Fig. 196 is a typical

°A'B'C'DIEIF

Time

FIG. 196. — VARIATION OF LOAD FACTOR DURING FLIGHT.

Each of the intervals A, B, ... is 30 sec. A, bad atmospheric bumps ; B,
loop ; C, spin ; D, dive and flatten out ; E and F, mock fighting.

record for acrobatic flight, with the added zest of mock fighting ;
unit spacing in the vertical scale represents the maximum deflection
of the wire under its own weight, i.e. due to the acceleration g.
Even in this fairly severe test, the pilot kept C.G. accelerations

528 AERODYNAMICS [CH. XII

between 0 and 4g. A specially trained pilot may nerve himself to
7g in cornering during a race, but such occasions are very exceptional,
and it appears that in ordinary flight the variation of load factor, as
due to control, is much less than might be expected. The accelera-
tions of parts far from the C.G. vary through a wider range. It
should be noticed that the accelerometer can be arranged to measure
these, and also the component acceleration in any desired direction.
The response of a stable aeroplane to disturbance creates stresses,
but the uncontrollable load factors may be more severe if the craft
be unstable. A growing upgust, for example, will generate excess
lift as indicated in Article 177, and this will be all the more marked
if the craft noses away from the relative wind.

* * * * *

To trace reliably the load factors arising from a gust of given
structure, as also to investigate many problems of control, we require
to follow the disturbed motion of the craft. Problems frequently
arising are concerned with the effects, over a comparatively brief
interval of time, of forces and moments suddenly applied, e.g.. by
dropping a load by parachute, or, in military aeronautics, by firing
a gun of considerable calibre. Others, as the analysis of spinning,
involve longer periods Such studies call for wider reading than the
foregoing treatment. Representation of the general motion of an
aeroplane following small disturbances leads to eight simultaneous
linear differential equations, though in some instances a smaller
number will suffice. The labour of solution is greatly reduced by
use of the method of operators introduced by Heaviside,* the theory
of which has been given by Jeffreys, t Applications of operational
methods to aircraft have been described by Bryant and Williams, J
Klemin,§ and others.

* Proc. Roy. Soc., A, 1893-4.

t Operational Methods in Mathematical Physics, 1927.

J A.R.C.R. & M., 1346, 1930.

§ Jour. Aeronautical Sciences, May 1936.

AUTHOR INDEX

The numbers refer to pages.

Abbot, 406
Ackeret, J., 264
Aihara, T., 253

Bailey, A., 106

Bairstow, L., 119, 369, 470, 493

Baker, G. S., 390

Beavan, J. A., 106

Bell, A. H., 102

Bernoulli, D., 35, 252

Betz, A., 88, 224, 323

Bioletti, C., 450

Blasius, H., 352, 373, 390, 394

Boss, N. K., 334

Boussinesq, J., 196, 401

Bradfield, Miss F. B., 89

Bryan, G. H., 119, 493

Bryant, L. W., 498, 528

Burgers, J. M., 341, 386

Busemann, A., 266

Cave, Miss B. M., 369
Cierva, J. de la, 446
Clark, K. W., 89
Cope, W, F., 6
Cowley, W. L., 504

Diehl, W. S., 454
Douglas, G. P., 442, 443
Dryden, H. L., 412
Drzewieki, S., 430

Eiffel, G., 68
Euler, L., 247

Fage, A., 211. 298, 353, 384, 394, 397,

409, 410, 423
Fairey, Sir R., 476
Fairthorne, R. A., 89
Falkner, V. M., 211, 390, 393, 394, 397
Farren, W. S., 82
Froude, W. and R. E., 65, 425

Garrard, W. C., 238, 415

Garner, H. M., 524

Gates, S. B., 516

Gebers, 390

Glauert, H., 174, 211, 234, 242, 259, 320,

339, 341, 354, 379, 440, 446, 493
Glauert, Mrs. H., 221, 234
Goett, 88

Goldstein, S., 108, 237, 415, 441
Gouge, A., 456, 473
Gray, W. E., 386

Hagen, G., 350
Handley Page, Sir F., 161
Hansen, M., 385
Hartshorn, A. S., 341, 442
Helmholtz, H. v., 270, 289
Hiemenz, K , 394
Hocker, C., 405
Hooker, S. G., 263, 266
Hooper, M. S., 354
Houghton, R., 6
Howarth, L., 393, 421
Hugoniot, 116
Hyde, G. A., 106

Imai, I., 253

Jeans, Sir J. H., 362

Jeffreys, H., 528

Jennings, W. G., 422

Jones, Sir B. M., 87, 397, 459, 493

Jones, E. T., 422

Jones, R., 102. 397, 410

Jouguet, E., 264

Joukowski, N., 203, 228

Kaplan, C., 253
Karden, H., 300
Karman, Th. v., 210, 253, 291, 341,

387, 390, 402

Kelvin, Lord, 192, 248, 270
Keinpf, G., 390
Kerber, L. V., 461

629

530

Keune, 224
Klemin, A., 528
Kutta, W., 221

Lachmann, G. V., 498

Lamb, Sir H., 237, 286, 287

Lanchester, F. W., 68, 118, 430, 493, 501

Lang, Miss, 369

Langley, 68

Lavender, T., 397, 623

Lees, C. H., 352, 357

Lesley, E. P., 470

Levy, H., 604

Lindsey, W. F., 262

Lilienthal, O., 68

Liptrot, R. N., 473

Littell, R. E., 262

Lock, C. N. H., 106, 430, 446

Maas, Van der, 452
Maccoll, J. W., 269
Marshall, Miss D., 385
Maxwell, J. C., 29, 32
Mines, R., 44
Mises, R. v., 234
Munk, M., 328, 498

Nagamiya, T., 474
Nazir, P. P., 162, 514
Nikuradse, 392

AUTHOR INDEX

Oseen, C. W., 370
Otten, G., 452
Ower, E., 523

Pannell, J. R., 351

Perring, W. G. A., 442

Phillips, H., 118

Piercy, N. A. V., 44, 53, 151, 212, 219,

290, 301, 354, 362, 371, 373, 393, 395,

415

Piper, R. W., 212
Pohlhausen, K., 394
Poiseuille, J. L. M., 350
Prandtl, L., 88, 266, 289, 300, 330, 334,

370, 387, 392, 395, 401, 405, 420, 440
Prescott, J., 362
Preston, J. H., 212, 219, 373, 393

Rankine, W. J. M., 116, 188, 267, 426
Rayleigh, Lord, 32, 58, 117, 188, 253, 367
Reid, E. G., 470

Relf, E. F., 102, 118, 276, 397, 405, 413,

623

Reynolds, O., 60, 350, 399
Riaboushinsky, D., 254
Richardson, E. G., 290, 369, 379, 395
Rolinson, D., 149
Routh, E. J., 280, 505
Rubach, 291

Saph, 351

Schlichting, H., 405

Schoder, 351

Sharman, C. F., 254

Simmons, L. F. G., 298

Squire, 395, 410

Stack, J., 262

Stanton, Sir T., 261, 351, 404

Stokes, Sir G., 370

Takenouti, Y., 474

Taylor, Sir G., 108, 254, 259, 266, 368,

402, 422

Tchapliguine, A., 263
Terazawa, K., 336
Thomson, Sir W., see Kelvin
Theodorsen, T., 237
Tietjens, O. G., 232, 350
Tomotika, S., 474
Townend, H. C. H., 353, 452
Trefftz, E., 210
Tsien, H., 263
Tyler, R. A., 393

Walker, W. S., 211, 397

Weick, R. E., 422

Wheatley, J. B., 460

White, C. M., 355

Whitehead, L. G., 212, 219, 393, 415,

423, 424

Wieselsberger, C., 390
Williams, D. H., 397, 410, 528
Winny, H. R, 354, 371, 386
Wood, S. A., 106
Wood, 422

Wright Brothers, 119
Wright, D., 135
Wright, K. V., 524

Young, A. D., 89, 395, 396

Zijnen, 386

SUBJECT INDEX

The numbers refer to pages.

Acceleration from rest, of flow, 191

of aerofoil, 296
Accelerometer, 527
Ackeret's theory, 264
Actuator, 425
Acyclic flow, 171
Adiabatic flow, 37
Aerodynamic centre, 92, 243, 497

climb of airship, 129

efficiency, 477

force, 54, 75
coefficient, 76

scale, 61, 67, 97

smoothness, 404
Aerofoil characteristics, preliminary, 89

shaping, 207, 210, 215, 219, 225, 414,
417, 423

testing, 80, 106, 335, 413

theory, 194, 237, 259, 309, 412, 424

velocity curves, 416
Aerofoils, Joukowski, 203, 221

Karman-Trefftz, 210, 219, 224

laminar flow, 412

Piercy, 212, 219, 225, 413

rectangular and tapered, 323
Ailerons, 119, 158, 161, 418, 624
Air brakes, 136
Air, properties of, 1
Aircraft, types of, 9, 118, 446
Airscrew factors, 434

interference, 461

slipstream effects, 465, 471, 481, 486
Airscrews, approximations for, 443

blade element theory, 429

compressibility losses, 442

practical formulae, 435, 444
, Speed, 471

static thrust, 439

tip losses, 440

variable pitch, 139, 438

vortex theory, 432
Airships, 9, 18, 122, 126, 478
Airspeed indicator, 43
Altimeter, 17, 490
Altitude effects, 17, 22, 67, 143, 466,

476, 484

Arbitrary wing, 320
Aspect ratio, 89, 302, 316, 319, 333,

337, 486

Atmosphere, standard, 16

isothermal, 13
Atmospheric tunnel, 71
Autogyro, 446, 487
Autorotation, 160

Balance, aerodynamic, 82

Balloons, 8, 18

Bank, angle of, 167

Bernoulli's equation, 35, 40, 50, 163,

186, 251, 426, etc.
Biplane, 123

definitions, 327

factor, 330, 454

of least drag, 331
Biplanes, tail planes of, 345

theorems relating to, 328
Blade angle, 428
Blockage, 108

Boundary, 26, 163, 187, 237, 261, 265,
364

layer, 52, 219; see flat plate
control, 419
theory, 370 et seq.
Breakaway, 178, 190, 219, 290, 393, 396,

409, 412, 419, 422
Buoyancy, 8, 18

Camber, 221, 224, 226, 237, 418, 423,

443

Cascade wing, 421
Cavitation, 170, 269
Ceiling, 19, 144, 469, 483
Centre of pressure coefficient, 80

of aerofoil, 90, 236, 237, 243, 418
of aeroplane, 154
of bulkhead, 11
stationary, 234, 244
travel, 91
Chattock gauge, 6
Circling flight, 156
Circular arc skeleton, 220, 226, 240

cylinder, 176, 195, 263, 292, 394, 397
Circulation and lift, 179, 230, 259
definition of, 49
generation of, 296
irrotational, 169, 228, 260, 366
persistence of, 248

631

532

SUBJECT INDEX

Circulation and lift — contd
prediction of, 228, 421
round wing, 228, 239, 260, 300, 310,

314, 323, 422
viscous, 365, 367

Climb, 129, 139, 438, 469, 472, 479, 488
correction for atmosphere, 491
slipstream, 472, 481
speed, 140
wind, 148
diagram, 492
Complex variable, 180

velocity, 181

Compressed-air tunnel, 96, 105
Compressible flow, 37, 103, 245, 259, 423

analogies, 254
Compressibility effects, 41, 108, 113,

253, 261, 266, 442
stall, see shock stall
Condensation, 257
Conformal transformation, 194

applied to aerofoils, 203, 210, 212,

221, 225

normal and inclined plate, 198
parallel walls, 280
Constraint, tunnel, 108, 113, 276, 304,

335

Continuity, equation of, 44, 166, 245
Controls, 119, 129, 133, 152, 158, 161,

623
Convergent flow, 85, 169, 413

nozzle, 72

Critical angle, 89, 135, 161
Mach number, 113, 262, 423
Reynolds number, 100, 350, 355
Curved flow, 367
Cyclic flow, 171
Cylinder in motion, 186

Damping factor, 495, 500, 503, 606, 612,

514
Density, definition, 1

of air, 2

of hydrogen and helium, 8

of water, 6

variation of in air flow, 37, 103, 116,

247, 253, 259
with altitude, 17, 67, 490
Derivatives, 505, 608, 614, 521
Descending flight, 146, 155, 158
Dihedral, longitudinal, 496

lateral, 162, 518
Displacement thickness, 391
Divergence, 496, 500
Divergent duct, 71, 185
Doublet, 172, 182, 285
Downwash, 149, 304, 339, 497, 609, 513
Drag, absence of in potential flow, 190

coefficients, 75

components of, 66, 364

form, 294, 410, 414, 418, 452, 455, 469

Drag, glider, 461, 471

induced, 123, 299, 311, 315, 328, 336,

453, 459

minimum, 126, 312, 331, 458, 478
of aerofoil, 90, etc.

aircraft parts, 466, 459, 464

airship, 123, 127, 398

autogyro, 449, 488

circular cylinder, 62

core in pipe, 366

flap, 134

flat plate, 99, 370, 385, 387, 398

normal plate, 136, 190

pipe, 350, 352, 403

rough plate, 406

sphere, 99, 409

streamline wire, 457

strut, 99, 398, 467
profile, 113, 409, 442, 445, 452
skin, 403

total parasitic, 123, 147, 486
Dynamical equations, Euler's, 247
Dynamic head, 41, 115
Dynamically similar motions, 60, 66,
75, 92, 96

Eccentric core in pipe, 356
Eddy viscosity, 400
Efficiency, Aerodynamic, 477

Jones, 459

of airscrew, 429, 437, 466

propulsion, 427, 477
Elasticity of air, 21
Electrical analogies, 264, 276
Elevator angle, 152

lag, 525
Elevators, 119

free, 516
Elliptic cylinder, 185, 188, 394, 422, 424

loading, 314, 322

wing, 315
Endurance, 476
Engine failure, 145

performance, 466, 471, 476, 485
Equal wing biplane, 332
Equipotentials, 164
Equivalence theorem, Munk's, 328
Equivalent monoplane aspect ratio, 463
Experiment, methods of, 61, 68, 304,
378, 386, 407, 509, 526

at high speeds, 103
Experimental mean pitch, 429

Fairing, 68

Fin, 119, 146, 454, 518
Fineness ratio, 176, 211, 467
Flaps, 133

Flat core in pipe, 356
' Flat plate ' glider drag, 452
Flat plate boundary layer, 372, 388, 392
friction, 373, 383, 386, 388

SUBJECT INDEX

633

Flow over faired nose, 172
in tunnel, 285

near stagnation point, 183

types of, 23
Form drag, see drag
Fourier series, use of, 243, 321
Frequency, 61, 292
Friction velocity, 402
Froude's law, 66

theory of propulsion, 425
Fuel consumption, 475

Gap, 327

Gas laws, 13, 15

Geometric pitch, 428

Glauert's dimensionless system, 505

lift theorem, 259
Glider drag, see drag

stability, 508
Gliders, 148
Gliding, 146

Handley Page slot, 161

Helical flight, 158

Helicopter, 120, 446, 487

High speeds, see Mach, pitot, subsonic,

supersonic, etc.
Hot wire anemometry, 43, 379
Hydraulic analogy, 254

mean depth, 355
Hydrostatic equation, 5
Hyperbola, inversion of, 212, 225
Hyperbolic channel, 186

Images, method of, 276, 338, 342
Impulse, 191, 287, 293, 298, 312
Impulsive pressure, 165
Incidence, 89, 229, 311, 316, 325

effect on laminar flow, 415
Indicated air speed, 44
Induced drag, see drag
method, 147

flow tunnel, 107

velocity, 270

Integration of Euler's equations, 250
Interference, 77, 108, 132, 152, 327

et seq., 399, 451, 452, 455, 457 et seq.
Isothermal flow, 37

Jet constraint, 338

propulsion, 142, 452, 484
Joukowski transformation formula, 203

aerofoils, 204, 221

approximate formulae for, 206
Joukowski's hypothesis, 227, 236, 297

Karman's boundary layer theorem, 380

modified form of, 392
similarity theory, 402
Karman trail, 292
Karman-Trefftz aerofoils, 210, 224

approximate formulae for, 211
Kelvin's theorem, 248
Kinematic coefficient of viscosity, 59, 67
Kinetic energy of irrotational flow, 192

minimum, 193, 312

of slipstream, 427

of trailing vortices, 299

Laminar flow, 23, 29, 49, 347, 354, 359,

373, 392
wings, 412
sub-layer, 353
Landing conditions, 132

run, 473

Laplace's equation, 166
Lapse rate, 17
Lateral oscillation, 519, 521

stability, 518
Lift, aerodynamic, of airship, 128

and circulation, 179, 230, 259, 296, 311
coefficients, 75

favourable range of, 415
curve slope, 233, 236, 261, 266, 317
evaluation from aerofoil pressures, 66,

109, 364

from trailing vortices, 301
from wall pressures, 109, 283
-drag balance, 79

ratio, 57, 75, 90, 92, 101, 266, 477,

488

elliptic, 314
generation of, 296
-grading, 323

effect on rolling moment, 523
of aerofoils, 90, 93, 102, 131, 133, 232,

236, 311, 315, 321
at subsonic speeds, 114, 259, 442
at supersonic speeds, 114, 264
of autogyro, 449, 488
of spinning cylinder, 180
of elliptic cylinder, 422
static, 8, 10, 18
uniform, 300, 324
Load, disposable, 122, 145

factor, 526

Longitudinal stability, 494, 496
graphical analysis, 516

Mach number, 64, 103, 423

angle, 262
Maxwell's law, 31
Maximum velocity ratio, 220, 417, 423

thickness location, 206, 210, 215, 219,
414, 418

534

SUBJECT INDEX

Mean camber, 224

motion, equations of, 399
Minimum flying speed, 131, 488
Mixing length, Prandtl's, 401
Moment, pitching, 92, 119, 152, 233,
239, 418, 495

coefficients of, 75

rolling, 158, 162, 495, 522

yawing, 146, 168, 495, 623
Momentum thickness, 391
Monoplane theory, equations of, 311

Non-dimensional coefficients, 76, 505
Normal plate, 136, 189, 193, 198, 313

profile drag, 409
Nose dive, 155

Open jet tunnel, 74, 97

constraint, 338
Operational methods, 528
Orthogonal biplane, 327
Oscillation, see phugoid, short, etc.
Oseen's approximation, 369

for flat plate, 370
Oval cylinder, 1 75

Parachute, 57
Parallel flow, see laminar
Path lines, 24, 189, 274, 282, 287
Parasite drag, 77, 137, 456, 460
Performance, prediction of, 451, 466,
471, 482

reduction of, to standard conditions,

148, 489

Phugoid oscillation, 501, 507, 512, 525
Piercy aerofoils, 207, 212, 219, 225, 410,
413

approximate formulae for, 215, 227
Pipe flow, steady, 349, 354

turbulent, 351, 355
Pitch, of aircraft, 119, 495

of airscrew, 428

variable, 139, 438
Pitot boundary, 53

head, 41

and vorticity, 51

at supersonic speed, 114

-static tube, 43

tube, 41

fractional, 379
Potential flow, 166, 250

function, 180

temperature, 21
Power curves, 137, 143, 157, 468

dive, 146

factor of tunnel, 71, 103

formulae, 387

loading, 127, 459, 480, 482

Prandtl's approximation, 370, 385
Pressure diagrams, aerofoil, 65, 91, 219

airship, 127

circular cylinder, 178

strut, 219
static, 2, 17, 34
variation in flow, 39, 115
Profile drag, see drag
Propeller, ideal, 425
Pusher airscrew, 462, 464

Qualitative compressibility effect, 253
theory of viscosity, 27

Range, 475

Rankine-Hugoniot law, 116
Rankine's method, 187

vortex, 267

Rarefied air tunnel, 104
Rayleigh's formula, 68
Rectangular aerofoil, 323
Reduction formulae, 317
Relative density, of air, 17, 44

of aeroplane, 496, 506
Reynolds number, 60, 97, 104, etc.

' effective,' 408

transition, 384, 386, 389
Rolling subsidence, 518, 621
Rotating cylinder, 365
Roughness, 403, 406
Routh's discriminant, 505, 517
Rudder, 119, 157, 518, 523

Scale effect, 62, 93, 98, 101, 385, 398,

407, 410

' Second problem, aerofoil theory,' 312
Sesqiii-plane, 327
Seventh-root law, 352, 387
Shock stall, 113, 262, 442

wave, 106, 112, 115, 262, 266, 423
Short oscillation, 498, 507, 512, 625
Singular points, 198, 216, 286
Sink, 168
Slip, absence of, 25, 350

of airscrew, 429

Slipstream, 425, 463, 465, 472, 480
Skin drag, 403

friction, 31, 55, 363

coefficient, 397

distribution on monoplane, 459

measurement of, 378

of aerofoils, 398, 410

of airships, 398

of cylinders, 393

of plates, see flat plate

of pipes, see pipe

reduction of, on wings, 411

transitional, 389

SUBJECT INDEX

535

Smoothness, aerodynamic, 405, 406
Smooth-turbulent formulae, 404
Soaring, 149
Solidity, of airscrew, 427, 434

of rotor, 488

Sonic throat, 38, 106, 110
Sound, velocity of, 22

wave, 259, 263, 265
Source, 167, 285

and sink, 171
Span-grading, 323

-loading, 315

Specific consumption, 476
Spin, 74, 160, 526
Spinning divergence, 521
Spoilers, 137, 524
Stability, asymmetric, see lateral

atmospheric, 20

coefficients, 504, 507, 511, 520

graphical analysis, 153, 497, 516

longitudinal, see longitudinal

of fluid motion, 292, 355, 367, 418
Stagger, 327
Stagnation point, 41, 183, 227, 368, 395

pressure, 42, 115

Stalling, 89, 102, 131, 135, 160, 290,
301, 420, 422, 438, 440, 449, 488

of Controls, 75, 153, 161, 418, 524
Starting vortex, 296
Stilling length, 351
Stoke's approximation, 369

operator, 247
Stratosphere, 1, 14, 21, 22, 63, 67,

485

Stream function denned, 47
Streamline, 23

aeroplane, 459

wall, 107

wires, 457

Stresses, component, 32, 360, 400
Strip method, 159, 430, 522
Strut, 99, 210, 219, 442, 457
Subsidence, 496, 500
Subsonic flow, 63, 250, 259, 423

experiment in, 103, 113, 442
Successive approximation method, 373
Supersonic flow, 114, 115, 262 et scq.
experiment in, 109, 113, 261, 166
Surface of discontinuity, 238, 288
Symmetric flight, 122, 494

Tail angle, 210, 214

efficiency, 513

-first lay-out, 498, 513

plane, 119, 149, 339, 497, 500, 509

-setting angle, 152, 154, 497
Take-off, 134, 136, 149, 438, 473
Tank, electric, 254, 276

ship, 66

Tapered wing, 323, 326
Temperature variation in flow, 37
Thomson's theorem, see Kelvin
Thrust and torque coefficients, 435, 440,
443, 467, 471

apparent, 462
Townend ring, 452
Trailing vortex pair, 297 et seq.
Transformation formula?, 194, 198, 203,

210, 213, 216, 281
Transformed sections, 204, 207, 210,

219, 226, 230, 417, 419
Transition, 385, 398, 409

curve, 389, 398

delay of, 412, 413, 419
effect on friction, 389, 411
effect on form drag, 455
effect 011 profile drag, 413, 486
effect of incidence, 416

detection, 386

point, 386

Reynolds number, 386, 472, 481

tunnel and flight, 399, 407, 410
Troposphere, 1, 14
Tunnel, atmospheric, 71

compressed air, 96

constraint, 108, 276, 281, 335

full-scale or giant, 74, 98

high-speed, 103, 106

laminar flow, 413, 455

supersonic, 109

variable-density, 103

vertical spinning, 74
Turbulence factor, 407

gauge of, in tunnel, 101
Turbulent flow, 24, 99, 351, 387, 395,

309 et scq., 407, 409 et scq.

Uniform flow, 23, 181

lift, 300

Upgust, 319, 528
Upward wind, 148

Velocity amplitude, 368, 395

defect, 403

induced, 270, 272, 310, 329

in potential flow, 183, 208

-potential, 163

physical explanation of, 165

ratio diagram, 416

maximum, 220, 423
Venturi tube, 44, 186
Viscosity, coefficient of, 31

theory and laws of, 27-31

eddy, 400

636

SUBJECT INDEX

Viscous flow, equations of, 362
Vortex, between walls, 281

bound, 272, 283

laws of in viscid, 270

pair, 274, 278
energy of, 290
impulse of, 287

Rankine's, 268

sheet, 288

starting, 296

street, 291

theory of airscrews, 432
Vortices, generation of, 287

trailing, 298 et scq.

wing-tip, 300
Vorticity, 49-51, 237, 245, 250, 267,

298, 349, 359, 364, 371, 402

Wake blockage, 108

effect on pressures, 393
on tail plane, 513

exploration, 86, 109
Wall, flexible tunnel, 107
Wave, see shock, sound

-making resistance, 65
Waves in water channel, 256
Wind axes, 495

effects on flight, 148

-tunnel corrections, 84, 169, 335, 344
Wing-loading, 122, 131, 136, 479, 484

Yaw, 119, 146, 156, 162, 454, 455, 495,
518, 523

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